Introduction

Charge density wave (CDW), which refers to a charge-ordered phase with a periodic modulation of charge density, plays a crucial role in understanding electron correlations, electron–phonon coupling, unconventional superconductivities, and quantum phase transitions1,2. However, it is difficult to predict critical temperatures of CDW transitions correctly using current theoretical and first-principles computational methods3, due to uncertain formation mechanisms, including Peierls instability, Fermi surface nesting, Kohn anomaly, excitonic insulator, or band Jahn–Teller effect4,5.

Phase transitions with static atomic displacement can be rationalized from a microscopic perspective by the soft mode theory (SMT)6. Unlike Landau’s theory, which describes phase transitions with order parameters from the view of spontaneous symmetry breaking7,8, SMT determines phase transitions by the soft modes of corresponding elementary excitation9. For example, order–disorder phase transitions can be explained by SMT of pseudospin waves10. The displacive type of ferroelectrics can be understood by SMT of optical phonons11. Dissimilarly, CDWs can be interpreted by SMT of acoustic phonons4, because the acoustic modes with isotropic vibration of atoms do not change the polarization intensity.

The key to determining CDW transitions based on SMT is to calculate the temperature-dependent phonon dispersions with phonon softening12 and imaginary part of phonon frequencies vanishing phenomena3. Many methods have been developed to calculate phonon dispersions at finite temperatures which must take into account phonon–phonon interactions or phonon anharmonicity13,14,15,16,17, such as temperature-dependent effective potential method18, self-consistent phonon theory19, and stochastic self-consistent harmonic approximation method20. Nevertheless, using above methods to compute the temperature-dependent phonon dispersions involves very large costs, and requires to adjustment of certain parameters with reference to experimental results, especially the smearing parameter σ15. A simpler approach is to simulate the temperature effect of the electronic system by directly adjusting the smearing parameter σ. Because the smearing method can only simulate the excited state of electrons system, its corresponding temperature Te = σ/kB will be much higher than the crystal temperature Te-p, which is an electron–phonon coupled system21.

In fact, the issue of excessively high electronic temperatures using smearing has been reported before. An order of magnitude difference was found between the critical temperature predicted by smearing and the experiment in 2H-NbSe23. Moreover, Duong et al.22 have discussed the effects of smearing methods on calculated critical temperatures of CDWs for bulk TiSe2. The Methfessel–Paxton (MP) smearing would result in a much higher critical temperature (around 3160 K) than the Fermi–Dirac smearing (around 500 K) and experimental results (around 200 K)22. Despite the significant differences, temperature effects induced phase transition and phonon softening phenomena simulated by smearing methods at Te agree with the behavior of a actual crystal system at Te-p3. Therefore, there may be some correlations between the smearing and actual temperatures. The electronic critical temperature Te,c using smearing does need to be corrected, but currently there has been no good solution for this correction.

In this paper, a three-temperature model was developed to rescale the smearing temperature Te and determine the critical temperature of CDWs without finite temperature corrections, where soft-mode phonons were assigned an independent temperature to describe their role in the energy transport process. Eight different layered materials TX2 (Table 1) were used to test the proposed method, including single charge density wavevector system (monolayer 1T-TiSe2, monolayer 2H-NbSe2, monolayer 2H-TaS2) and multiple wavevectors system (bulk 1T-TiSe2, bulk 1T-TaS2, bulk 2H-TaSe2, bulk 2H-TaS2, bulk 2H-NbSe2), low (bulk 2H-NbSe2, bulk 2H-TaS2, bulk 2H-TaS2, monolayer 2H-TaS2), and high (monolayer 1T-TiSe2, bulk 1T-TiSe2, bulk 1T-TaS2) critical temperature materials.

Table 1 Critical temperatures of TX2
Full size table

Results

Mode-selective smearing

Assuming that the electronic band structure εi at zero temperature has been calculated, the objective is to analyze the impact of changing the electronic temperature on the phonon spectrum calculation using the smearing method. Within the Born–Oppenheimer approximation23, the lattice dynamical properties of the system are governed by the Schrödinger equation24:

$$\left[-\sum _{I}\frac{{\hslash }^{2}}{2{M}_{I}}\frac{{\partial }^{2}}{\partial {{\bf{R}}}_{I}^{2}}+{E}_{{\rm{N}}}\left(\left\{{{\bf{R}}}_{I}\right\}\right)+{E}_{\left\{{{\bf{R}}}_{I}\right\}}\right]\chi \left(\left\{{{\bf{R}}}_{I}\right\}\right)={\mathcal{E}}\chi \left(\left\{{{\bf{R}}}_{I}\right\}\right).$$
(1)

Here \({\mathcal{E}}\) is the total energy of the system, \(\chi \left(\left\{{{\bf{R}}}_{I}\right\}\right)\) is the nuclear wavefunction, \({E}_{{\rm{N}}}\left(\left\{{{\bf{R}}}_{I}\right\}\right)={e}^{2}/2{\sum }_{I\ne J}{Z}_{I}{Z}_{J}/\left\vert {{\bf{R}}}_{I}-{{\bf{R}}}_{J}\right\vert\) denotes the Coulomb repulsion between different nuclei, and \({E}_{\left\{{{\bf{R}}}_{I}\right\}}\) is the total electronic energy at clamped nuclei calculated from the density-functional theory, where \(\left\{{{\bf{R}}}_{I}\right\}\) are parameters as the nuclei generate a static external potential. \({E}_{\left\{{{\bf{R}}}_{I}\right\}}\) can be written as24

$$\begin{array}{ll}{E}_{\left\{{{\bf{R}}}_{I}\right\}}={E}_{band}-\frac{{e}^{2}}{2}\int\frac{n({\bf{r}})n\left({{\bf{r}}}^{{\prime} }\right)}{\left\vert {\bf{r}}-{{\bf{r}}}^{{\prime} }\right\vert }d{\bf{r}}d{{\bf{r}}}^{{\prime} }+{E}_{{\rm{xc}}}[n]\\\qquad\quad -\int{V}_{{\rm{xc}}}({\bf{r}})n({\bf{r}})d{\bf{r}}.\end{array}$$
(2)

Here, the single-particle energy Eband and the electronic density n(r) can be expressed as

$${E}_{band}=\mathop{\sum }\limits_{i=1}^{N/2}{\varepsilon }_{i},$$
(3)
$$n({\bf{r}})=\mathop{\sum }\limits_{i=1}^{N/2}{\left\vert {\psi }_{i}({\bf{r}})\right\vert }^{2}.$$
(4)

The single-particle energy eigenvalues εi and orbitals ψi(r) are auxiliary quantities derived by self-consistently solving the Kohn–Sham equations25. Furthermore, ψi(r) depends only parametrically on the nuclear coordinates RI.

When smearing is introduced, the single-particle energy and the electronic density become

$${E}_{band}^{s}=\mathop{\sum }\limits_{i=1}^{N}{f}_{i}{\varepsilon }_{i},$$
(5)
$${n}^{s}({\bf{r}})=\mathop{\sum }\limits_{i=1}^{N}{f}_{i}{\left\vert {\psi }_{i}({\bf{r}})\right\vert }^{2}.$$
(6)

Here, fi is the occupation function which is related to the smearing methods (details are provided in the “Methods” section). Hence, when using smearing methods, the total electronic energy at clamped nuclei can be written as

$$\begin{array}{ll}{E}_{\left\{{{\bf{R}}}_{I}\right\}}^{s}={E}_{band}^{s}-\frac{{e}^{2}}{2}\int\frac{{n}^{s}({\bf{r}}){n}^{s}\left({{\bf{r}}}^{{\prime} }\right)}{\left\vert {\bf{r}}-{{\bf{r}}}^{{\prime} }\right\vert }d{\bf{r}}d{{\bf{r}}}^{{\prime} }+{E}_{{\rm{xc}}}[{n}^{s}]\\\qquad\quad -\int{V}_{{\rm{xc}}}({\bf{r}}){n}^{s}({\bf{r}})d{\bf{r}}.\end{array}$$
(7)

The total potential energy of the nuclei is given by \({U}^{s}={E}_{{\rm{N}}}\left(\left\{{{\bf{R}}}_{I}\right\}\right)+{E}_{\left\{{{\bf{R}}}_{I}\right\}}^{S}\), which is also referred to as the Born–Oppenheimer energy surface. Hence, the force constant matrix is

$${\Phi }_{I,J}=\frac{{\partial }^{2}{U}^{s}}{\partial {R}_{I}\partial {R}_{J}}=\frac{{\partial }^{2}{E}_{{\rm{N}}}\left(\left\{{{\bf{R}}}_{I}\right\}\right)}{\partial {R}_{I}\partial {R}_{J}}+\frac{{\partial }^{2}{E}_{\left\{{{\bf{R}}}_{I}\right\}}^{s}}{\partial {R}_{I}\partial {R}_{J}}.$$
(8)

In the first term of Eq. (8), which represents the Coulombic interactions between ions, the smearing effect is not involved. The second term consists of three parts: the first part corresponds to the single-particle energies, which are directly influenced by the smearing effect by altering the electronic filling function. The remaining three parts are related to the electron density n(r) and also experience an indirect influence from the smearing, which can be revealed from the differential electron charge distribution for two smearing parameters, Δρ = ρ(σ = 0.098 eV) − ρ(σ = 0.070 eV), shown in Fig. 1a. In turn, the force constant matrix could be changed by smearing, which affects the calculation of phonon frequencies. In particular, phonon modes whose potential energy surfaces contributed mainly by electronic states close to the Fermi level, would experience significant corrections to their frequencies, as their occupation numbers are more sensitive to smearing parameters. In contrast, smearing has little influence for phonon modes whose potential energy surfaces are contributed mainly by electronic states with energies much lower and higher than the Fermi level. Therefore, the calculated temperature-dependent phonon dispersions using smearing present two different phonon modes, i.e., the soft modes and non-soft modes.

Fig. 1: The smearing effect on the charge density and force constant matrix.
figure 1

a is the top and side views of the positive and negative differential electron charge distribution of monolayer 1T-TiSe2 resulting from two smearing parameters, Δρ = ρ(σ = 0.098 eV) − ρ(σ = 0.070 eV). b Potential energy surfaces of monolayer 1T-TiSe2 with different smearing temperature Te for the soft-mode and non-soft-mode phonons. c Temperature change schematics of electrons, soft-mode and non-soft-mode phonons in the three-temperature model, where τe-sp and τsp-nsp are relaxation time of electrons and soft-mode phonons, soft-mode, and non-soft-mode phonons, respectively. The effect of smearing is to heat electrons to a high temperature Te. When time t < τe-sp, electrons excite more soft-mode phonons through mode-selective smearing. When time t > τe-sp, electrons and soft-mode phonons are in thermal equilibrium, and soft-mode phonons excite other non-soft-mode phonons through introduced phonon–phonon scattering described by Gsp-nsp. When time t > τsp−nsp, electrons and phonons are in thermal equilibrium.

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This kind of temperature effects can be revealed from the potential energy surface shown in Fig. 1b. There are significant differences for soft-mode phonons between temperatures higher and lower than the critical temperature Te,c, where the former has one potential well but the latter has two. This means that there are more stable structures with lower energy, i.e., the CDW phase, than the normal phase. Nonetheless, the potential energy surface for non-soft-mode phonons remains unchanged. Therefore, the smearing method is also mode-selective, where the soft-mode phonons are selected by the electronic contribution of the force constant matrix which can be changed by smearing parameter. This effect is just like mode-selective electron–phonon coupling in CDWs26,27,28. In fact, temperature mainly influences the interaction between ions or phonons and electrons, as the temperature excites an increasing number of phonons, but with no phonon excitations at zero temperature. This is why the smearing method is not well suited for handling temperature effects, as its direct impact on phonons is much more significant than on electrons.

Considering the mode-selective impact of smearing, soft-mode phonons will be largely corrected. Nevertheless, smearing has almost no effect on non-soft-mode phonons as the ionic energy Eion does not change with smearing. Hence, only the soft-mode part is a high-temperature spectrum in temperature-dependent phonon dispersions calculated using smearing methods, whereas the non-soft-mode part is still a zero-temperature spectrum. From the transport perspective, energy can be transported by strong interaction of electrons near the Fermi-level and soft-mode phonons but with no energy change for non-soft-mode phonons.

Three-temperature model

Indeed, mode-dependent multi-temperature energy transport processes have been predicted and experimentally measured in ultrafast processes29,30. Given the non-adiabatic coupling between electrons and soft-mode phonons, a three-temperature model is developed for describing the energy transport between electrons, soft-mode phonons, and non-soft-mode phonons, based on the traditional two-temperature model31.

As shown in Fig. 1c, the choice of the smearing parameter σ is equivalent to contacting the electrons with a constant temperature heat source Te, so that the electron temperature is constant at Te. Then, the temperature of soft-mode phonons continues to rise until Tsp = Te = Te,c by the mode-selective effect of smearing, or in other words, more soft-mode phonons are excited by excitation of electrons near the Fermi level, when time t < τe-sp.

The reason why the critical temperature predicted by smearing methods is excessively high is that the electron temperature Te,c is mistakenly assumed to be the equilibrium temperature of the electron–phonon coupling system. In fact, most phonons are still in the ground state when electrons are at high temperatures. Therefore, the equilibrium temperature Te-p,c after electrons and phonons relaxation should be the critical temperature instead of Te,c, as shown in Fig. 1. Changes after the relaxation of electrons and soft-mode phonons should be further analyzed. When time t > τe,sp, electrons and soft-mode phonons are always in thermal equilibrium Tsp = Te, but the soft-mode phonons will excite other non-soft-mode phonons through phonon–phonon interaction. At this point, the temperature of non-soft-mode phonons increases and those of electrons and soft-mode phonons decrease until they reach a new thermal equilibrium Tnsp = Tsp = Te = Te-p,c, as shown in Fig. 1c. This new equilibrium temperature Te-p,c should be recognized as the critical temperature of CDWs.

To be more specific, the whole system is approximated as three subsystems: electron gas, soft-mode phonon gas, and non-soft-mode phonon gas. The electron gas and soft-mode phonon gas are in thermal equilibrium at a temperature Te, due to the mode-selective smearing effect. The non-soft-mode phonons, on the other hand, remain in their ground state, i.e., at Tnsp = 0 K, or in other words, no non-soft-mode phonons have been excited. The problem is to determine the temperature at which all three subsystems are in thermal equilibrium. When interactions are introduced, non-soft-mode phonons can only be excited by soft-mode phonons, since the mode-selective smearing effect has been taken into account. Therefore, this problem can be solved using a simple three-temperature model, just like the traditional two-temperature model that can be used to describe the energy transfer between electrons and phonons31. It is important to note that the new interactions introduced to achieve thermal equilibrium among the three subsystems will only affect the time required to reach equilibrium, but will not change the final equilibrium temperature, which is the concern of this study.

Without considering spatial heat transfer, the three-temperature model can be expressed as

$${C}_{e}({T}_{e})\frac{\partial {T}_{e}}{\partial t}=-{G}_{e{{\text{-}}sp}}\left({T}_{e}-{T}_{sp}\right),$$
(9)
$${C}_{nsp}({T}_{nsp})\frac{\partial {T}_{nsp}}{\partial t}=-{G}_{sp-nsp}\left({T}_{nsp}-{T}_{sp}\right),$$
(10)
$$\begin{array}{rcl}{C}_{sp}({T}_{sp})\frac{\partial {T}_{sp}}{\partial t}=-{G}_{e{{\text{-}}sp}}\left({T}_{sp}-{T}_{e}\right)-{G}_{sp{{\text{-}}nsp}}\left({T}_{sp}-{T}_{nsp}\right).\end{array}$$
(11)

Here Te, Tsp, and Tnsp are the temperatures of electrons, soft-mode phonons and non-soft-mode phonons, respectively. Ce, Csp, and Cnsp are the heat capacities of electrons, soft-mode phonons, and non-soft-mode phonons, respectively. Ge-sp and Gsp-nsp are the coupling coefficients of electrons and soft-mode phonons, soft-mode phonons, and non-soft-mode phonons, respectively. Considering these interactions, the temperature of each subsystem should be involved as shown in Fig. 1c.

Solving the three-temperature model accurately is difficult because it is necessary to calculate coupling coefficients between electrons, soft-mode phonons, and non-soft-mode phonons first. Actually, coupling coefficients determine the length of the relaxation time, while we are only concerned with the final equilibrium temperature. Hence, the coupling coefficients can be eliminated. Substituting Eqs. (9) and (10) into Eq. (11) yields

$${C}_{sp}({T}_{sp})\frac{\partial {T}_{sp}}{\partial t}=-{C}_{e}({T}_{e})\frac{\partial {T}_{e}}{\partial t}-{C}_{nsp}({T}_{nsp})\frac{\partial {T}_{nsp}}{\partial t}.$$
(12)

Integrating time from τe-sp to τsp-nsp yields that

$$\begin{array}{ll}\mathop{\int}\nolimits_{{T}_{e,c}}^{{T}_{e\text{-}p,c}}{C}_{sp}({T}_{sp}){\rm{d}}{T}_{sp}=-\mathop{\int}\nolimits_{{T}_{e,c}}^{{T}_{e{\text{-}p,c}}}{C}_{e}({T}_{e}){\rm{d}}{T}_{e}\\\qquad\qquad\qquad\qquad\quad\,\,-\mathop{\int}\nolimits_{{T}_{nsp,0}}^{{T}_{e{\text{-}p,c}}}{C}_{nsp}({T}_{nsp}){\rm{d}}{T}_{nsp},\end{array}$$
(13)

where Tnsp,0 → 0 K and Te,c is the critical electronic temperature. According to the definition of heat capacity C = (∂E/∂T)V where E is the single-particle energy, the energy transport between electrons, soft-mode phonons, and non-soft-mode phonons can be written as

$$\begin{array}{rcl}{E}_{e}({T}_{e{{\text{-}}p,c}})+{E}_{sp}({T}_{e{{\text{-}}p,c}})+{E}_{nsp}({T}_{e{{\text{-}}p,c}})\\={E}_{e}({T}_{e,c})+{E}_{sp}({T}_{e,c})+{E}_{nsp}({T}_{nsp,0}).\end{array}$$
(14)

The critical temperature Te-p,c can be determined by calculating zero points of energy change function with temperature, which can be written as

$$\begin{array}{ll}\Delta E({T}_{e{{\text{-}}p}})={E}_{e}({T}_{e{\text{-}}p})+{E}_{sp}({T}_{e{{\text{-}}p}})+{E}_{nsp}({T}_{e{{\text{-}}p}})\\\qquad\qquad\quad-{E}_{e}({T}_{e,c})-{E}_{sp}({T}_{e,c})-{E}_{nsp}({T}_{nsp,0})\\\quad\quad\quad\quad\,=\Delta {E}_{e}({T}_{e{{\text{-}}p}})+\Delta {E}_{sp}({T}_{e{{\text{-}p}}})+\Delta {E}_{nsp}({T}_{e{{\text{-}}p}}).\end{array}$$
(15)

The above equation implies that the total energy change is equal to the sum of the energy changes in the three subsystems: electrons, soft-mode phonons, and non-soft-mode phonons. Considering the single-particle energy, ΔE(Te-p) can be expressed as

$$\begin{array}{ll}\Delta E({T}_{e{{\text{-}}p}})=2\sum\limits_{{\bf{k}}}{\varepsilon }_{{\bf{k}}}\left[f\left(\frac{{\varepsilon }_{{\bf{k}}}-\mu }{{k}_{B}{T}_{e{\text{-}}p}}\right)-f\left(\frac{{\varepsilon }_{{\bf{k}}}-\mu }{{k}_{B}{T}_{e,c}}\right)\right]\\\qquad\qquad \quad+\sum\limits_{{{\bf{q}}}_{s}}\hslash {\omega }_{{{\bf{q}}}_{s}}\left[\frac{1}{\exp \left(\frac{\hslash {\omega }_{{{\bf{q}}}_{s}}}{{k}_{B}{T}_{e{{\text{-}}p}}}\right)-1}-\frac{1}{\exp \left(\frac{\hslash {\omega }_{{{\bf{q}}}_{s}}}{{k}_{B}{T}_{e,c}}\right)-1}\right]\\\quad\qquad\qquad +\sum\limits_{{{\bf{q}}}_{ns}}\frac{\hslash {\omega }_{{{\bf{q}}}_{ns}}}{\exp \left(\frac{\hslash {\omega }_{{{\bf{q}}}_{ns}}}{{k}_{B}{T}_{e{{\text{-}}p}}}\right)-1}.\end{array}$$
(16)

Here the first term in Eq. (16) corresponds to the change in electronic energy between the corrected temperature Te-p and the electronic CDW transition temperature Te,c, as defined by ΔEe(Te-p) in Eq. (15). The second term represents the energy change of the soft-mode phonons between the same two temperatures, while the third term accounts for the energy gained by the non-soft-mode phonons when heated from 0 K to Te-p. Additionally, the electron occupation function f is related to smearing methods, and μ is the chemical potential of electrons that can not be approximated as the Fermi energy EF due to the large range of temperature variations. The chemical potential μ at different temperatures can be determined by the conservation of the particle number, which can be expressed as

$${N}_{e}=2\sum _{{\varepsilon }_{{\bf{k}}}}u({E}_{F}-{\varepsilon }_{{\bf{k}}})=2\sum _{{\varepsilon }_{{\bf{k}}}}f\left(\frac{{\varepsilon }_{{\bf{k}}}-\mu (T)}{{k}_{B}{T}_{e}}\right),$$
(17)

where u(EF − εk) is the unit step function, i.e., u(EF − εk) = 1 when εk ≤ EF and u(EF − εk) = 0 when εk > EF.

CDW transitions can be determined by observing the imaginary part of phonon dispersions at qCDW approaching zero and the electronic critical temperatures shown in Supplementary Fig. S3a–f can be calculated by the corresponding smearing parameters, i.e., Te,c = σc/kB, which are much higher than the critical temperature measured in experiments (see Fig. 3a, d and Table 1), because electrons and phonons are not fully relaxed. The equilibrium temperature Te-p,c for electrons, soft-mode, and non-soft-mode phonons can be determined by the three-temperature model (see Eq. (16)). Before that, temperature-dependent chemical potential μ(T) should be calculated based on the electron energy bands and mode-selective smearing induced soft-mode phonons should be identified based on the phonon dispersions. Results of chemical potential μ(T) are shown in Supplementary Figs. S3g, h and S5c, calculated by the conservation of electrons number (see Eq. (17))32. However, soft-mode phonons would be difficult to select if no quantitative standard were referenced.

Considering that the degree to which soft-mode phonons are softened is not the same with different smearing parameters σ, an absolute standard would result in soft-mode phonons dependent on σ. This is not feasible because there is no criterion for the choice of σ itself. Here a relative standard based on the degree of softening is introduced as follows.

Standard for soft-mode phonons

In view of that imaginary frequencies can reflect the degree of softening effectively9, the average relative softening strength can be defined as

$$\overline{{P}_{s}}=\frac{1}{{N}_{{\bf{q}}}}\mathop{\sum }\limits_{{\bf{q}}^{\prime}}\frac{{\omega }_{{T}_{e}}({{\bf{q}}}^{{\prime} })}{{\omega }_{{T}_{e},max}},$$
(18)

where \({{\bf{q}}}^{{\prime} }\in \left\{{\bf{q}}| {\omega }_{{T}_{e}}({\bf{q}}) < 0\right\}\), Nq is the number of phonon modes, \({\omega }_{{T}_{e}}\) is the phonon frequency at Te, respectively. Phonons with wavevector \({{\bf{q}}}^{{\prime} }\) contain all the softened phonons. However, phonons with imaginary frequencies are only a part of soft-mode phonons that are strongly softened. Hence, the standard for soft-mode phonons should go beyond the average relative softening strength. Assuming that the proportionality coefficient between the softening threshold \(\overline{{P}_{s}}\) and the average relative softening strength \(\overline{{P}_{s}}\) is γ, soft-mode phonons with wavevector qs can be defined as

$$\frac{{\omega }_{{T}_{e,c}}({{\bf{q}}}_{s})}{{\omega }_{{T}_{e,c},max}}-\frac{{\omega }_{{T}_{e}}({{\bf{q}}}_{s})}{{\omega }_{{T}_{e},max}}\ge \gamma \overline{{P}_{s}}.$$
(19)

For monoatomic chains, the phonon frequency is related to atomic mass when the atomic force constant is approximately unchanged, i.e., ω0(q) ∝ m−1/2. Extending to polyatomic chains, the phonon frequency is mainly modulated by the generalized effective atomic mass \({m}^{* }=\prod _{n}{m}_{n}/\sum _{n}{m}_{n}\), i.e., \({\omega }_{0}({\bf{q}})\propto {{m}^{* }}^{-1/2}\). Therefore, the softening strength γ should also be related to the effective mass, and this relationship should be relatively robust, especially for materials with similar structures. By testing the transition temperatures of different materials against the soft-mode phonons standard, we found that an empirical formula, \({R}_{\gamma }={{R}_{{m}^{* }}}^{n}\) where n is a parameter, can be used to fit the soft-mode phonons standard coefficient ratios Rγ to generalized effective atomic mass ratios \({R}_{{m}^{* }}\) of different materials as shown in Figs. 3b and 5h, when the difference in atomic force constants is not significant.

Discussion

To validate the proposed method used to correct the critical temperature using smearing Te,c, eight van der Waals CDW materials were tested, which are bulk 1T-TiSe2, bulk 1T-TaS2, bulk 2H-TaSe2, bulk 2H-TaS2, bulk 2H-NbSe2, monolayer 1T-TiSe2, monolayer 2H-NbSe2 (so called 1H-NbSe2) and monolayer 2H-TaS2 (so called 1H-TaS2). A complete list is shown in Table 1 and the atomic structures are shown in Supplementary Fig. S2a. Electron energy bands are shown in Supplementary Figs. S2c–f, S5a, and S7, and temperature-dependent phonon dispersions using smearing methods are shown in Figs. 2 and 5a–c and Supplementary Fig. S5b. Further details of the computational procedures are described in the “Methods” section.

Fig. 2: Phonon dispersions with different smearing parameters σ and smearing methods.
figure 2

a–c are phonon dispersions with different smearing parameters σ using Fermi, Gaussian, and MP smearing methods, respectively. d–f are phonon dispersions with different smearing parameters σ using Fermi smearing methods of bulk 1T-TiSe2, monolayer 2H-NbSe2, and monolayer 2H-TaS2, respectively.

Full size image

It should be noted that the soft-mode screening standard \({R}_{\gamma }={{R}_{{m}^{* }}}^{n}\) proposed in this work is, in our view, applicable to CDW materials across different systems. However, the parameter n used in the standard needs to be fitted based on a group of structurally similar materials. For example, the three 1T-phase (bulk 1T-TiSe2, bulk 1T-TaS2, monolayer 1T-TiSe2) and two 1H-phase (monolayer 2H-NbSe2, monolayer 2H-TaS2) materials selected in this study are structurally similar. This is because, for two-dimensional (2D) materials with the 1T phase, the bulk form is essentially a stack of identical monolayers, which can be observed in Supplementary Fig. S2a. In contrast, for the 2H phase, the crystal structure of the bulk is not the same as that of the monolayer. In such cases, the three-temperature model remains valid, but the parameter n should be fitted using materials of the same type, e.g., bulk or bilayer materials with 2H stacking. Hence, three more 2H-phase materials (bulk 2H-TaSe2, bulk 2H-TaS2, bulk 2H-NbSe2) were further tested.

For the 1T-phase and 1H-phase structures, monolayer 1T-TiSe2 was used as a reference in view of its relatively consistent and well-established experimental values for its CDW transition temperature33,34,35. The soft mode standard coefficient \({\gamma }_{{\text{TiSe}}_{2},\text{1L}}\) was first fitted with the experimental critical temperature, and the parameter n ≈ 4 was obtained. Then, the soft-mode standard coefficients for the other three materials \({\gamma }_{{\text{NbSe}}_{2},\text{1L}}\), \({\gamma }_{{\text{TaS}}_{2},\text{1L}}\) and \({\gamma }_{{\text{TiSe}}_{2},\text{bulk}}\), were calculated using the empirical equation \({R}_{\gamma }={{R}_{{m}^{* }}}^{4}\). According to γ and the average relative softening strength \(\overline{{P}_{s}}\) with different smearing parameters σ, screened soft-mode phonons for 1T-phase and 1H-phase are shown in Fig. 4 and Supplementary Fig. S4. Specifically, the screened soft-mode phonons for 1T-TiSe2 are in good agreement with those determined by the phonon linewidth36. For the 2H-phase structures (bulk 2H-TaSe2, bulk 2H-TaS2, bulk 2H-NbSe2), soft modes screening standard \({R}_{\gamma }={{R}_{{m}^{* }}}^{n}\) was fitted with n ≈ −5 using bulk 2H-TaSe2 as a reference, as shown in Fig. 5h.Screened soft-mode phonons for 2H-phase are shown in Fig. 5d–g and Supplementary Fig. S8.

Then, the proposed three-temperature model can be used to calculate the equilibrium temperature after relaxation of electrons, soft-mode and non-soft-mode phonons. Figure 3c, Supplementary Fig. S6h, and Figs. 3d and 5i illustrate absolute values of energy change functions ∣ΔE(Te-p)∣ and corrected critical temperatures determined by the zero points of ∣ΔE(Te-p)∣ for TX2, respectively.

The use of different smearing methods leads not only to different temperature-dependent phonon dispersions shown in Fig. 2a–c but also to different transition temperatures shown in Fig. 3a. Electron transition temperatures Te,c calculated by the Gaussian and MP methods are about three and five times higher than the Fermi method, respectively, and the corrected transition temperatures are also about two times higher than the Fermi method. Even though Gaussian and MP smearing astringe better than Fermi, their occupation functions deviate from the actual Fermi–Dirac distribution at high temperatures32. In contrast, Fermi smearing has a good astringency at high temperatures. Hence, only the Fermi smearing is recommended to compute temperature-dependent phonon dispersions and predict critical temperatures, while other methods like Gaussian and MP are not recommended.37.

Fig. 3: Three-temperature model corrections to the critical temperatures calculated by smearing.
figure 3

a Electronic critical temperatures calculated by smearing according to Supplementary Fig. S3a–f. b Empirical formula for screening soft-mode phonons, where the selection parameter ratio depends on the generalized effective atomic mass ratio, defined as \({R}_{\gamma }={{R}_{{m}^{* }}}^{n}\) (with n = 4 for the 1T-phase and 1H-phase). c Zero points of energy change functions with temperature for different smearing parameters σ. d Critical temperatures of TX2 corrected by the three-temperature model for Fermi, Gaussian, and MP smearing. Here, the experimental data are from refs. 33,34,35,54,55,56,57,58.

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Figure 3d shows the critical temperatures corrected by the proposed three-temperature model Te-p,c at different electronic temperatures Te, where electrons, soft-mode and non-soft-mode phonons are in thermal equilibrium. Critical temperatures Te-p,c are not only consistent with the experimental results, but are also independent of electronic temperatures Te, except for some significant deviations near electronic critical temperatures Te,c (for example, σ = 0.090, 0.092 for 1T-TiSe2 using Fermi smearing). The reason is that soft-mode phonons are hard to be screened certainly when using the phonon dispersions near Te,c (see Eq. (19)), which can be observed form soft-mode phonons of monolayer and bulk 1T-TiSe2 in Fig. 4d, e and Supplementary Fig. S4k. For the other three materials, 1H-NbSe2, 1H-TaS2 and bulk 1T-TaS2, the screened soft-mode phonons are similar, even using dispersions at different electronic temperatures, reflected from Fig. 4g–l and Supplementary Figs. S4a–e and S6a–g. The stability of critical temperatures and the consistency of soft-mode phonons jointly demonstrate the superiority of the proposed relative standard for soft-mode phonons.

Fig. 4: Screened soft-mode phonons (red sphere) for monolayer 1T-TiSe2 and 1H-TaS2.
figure 4

a–f are screened soft-mode phonons (red sphere) for monolayer 1T-TiSe2 using different smearing parameters σ, respectively. g–l are screened soft-mode phonons (red sphere) for monolayer 1H-TaS2 using different smearing parameters σ, respectively.

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For 2H-phase materials, the calculated critical temperatures using the three-temperature model demonstrate good agreement with experimental data for bulk 2H-TaSe2, 2H-TaS2 (see Fig. 5i and Table 1). However, bulk 2H-NbSe2 exhibits a relatively large discrepancy. A similar trend is observed in the screening of soft-mode phonons, where the soft modes for 2H-TaSe2 and 2H-TaS2 are reasonably well-screened, as evidenced in Fig. 5d–g and Supplementary Fig. S8a–c. In contrast, for 2H-NbSe2, under the soft modes screening standard \({R}_{\gamma }={{R}_{{m}^{* }}}^{-5}\) adopted in this study, no soft-mode phonons satisfy the conditions. This discrepancy may be attributed to the ultra-low CDW transition temperature of 2H-NbSe2 and a relatively weak thermal smearing effect compared to other materials, as evidenced by its calculated electronic critical temperature of approximately 835 K which is about 25 times higher than the experimental value of 33 K. Moreover, bulk 2H-NbSe2 exhibits an unusual CDW behavior, where the phonon softening occurs over a broad range of wavevectors with obvious shifting with temperature change (see Fig. 5c), rather than forming a sharp cusp at a specific point (see Fig. 5a, b), indicating that the CDW wavevector is governed by the extended momentum dependence of electron–phonon coupling rather than Fermi surface nesting3. Since the smearing method effectively elevates the electronic system to a high temperature, the equilibrium temperature between electrons and phonons, even in the absence of soft-mode phonon involvement, is at least 75 K, as evidenced by the energy variation function shown in Supplementary Fig. S8d. Nevertheless, the three-temperature model proposed in this study still achieves notable progress; in particular, it yields a transition temperature (≈75 K) that is at least an order of magnitude lower than that obtained using the smearing method alone (≈835 K using Fermi smearing in this paper, ≈2000 K using Gaussian smearing3).

Fig. 5: Phonon dispersions, screened soft-mode phonons (red sphere), and corrected critical temperatures for bulk 2H-TaSe2, 2H-TaS2 and 2H-NbSe2.
figure 5

a–c are phonon dispersions using different smearing parameters σ for bulk 2H-TaSe2, 2H-TaS2, and 2H-NbSe2, respectively. d–g are screened soft-mode phonons (red sphere) using different smearing parameters σ for 2H-TaSe2. h Empirical formula for screening soft-mode phonons, where the selection parameter ratio depends on the generalized effective atomic mass ratio, defined as \({R}_{\gamma }={{R}_{{m}^{* }}}^{n}\) (with n = −5 for the 2H-phase).

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By removing data resulting from incorrect screening of soft-mode phonons (for example, σ = 0.090, 0.092 for 1T-TiSe2 using Fermi smearing), average critical temperatures Te-p,c shown in Table 1, agree with experimental results well. It can be clarified that the computational method with Fermi smearing based temperature-dependent phonon dispersions, combined with the three-temperature mode, can effectively and efficiently predict the critical temperature of CDW transitions.

Furthermore, for quasi-one-dimensional systems such as MX3 (M = Ti, Nb, Ta and X = Se, S, Te, Zr, Hf)38 and A0.3MoO3 (A = K, Rb, Tl)39, the same approach can potentially be extended, provided that the CDW transition is driven by well-characterized soft-mode phonon instabilities. However, in systems where phonon softening is not sharply localized, for instance when the distinction between soft and non-soft phonons is ambiguous or the phonon softening occurs over a broad momentum range, the parameter n in our soft-mode screening standard cannot be reliably determined, and consequently, the applicability of the three-temperature model becomes limited. This is because the model fundamentally relies on the mode-selective influence of smearing on soft-mode phonons, which enables the energy redistribution between electrons and phonons to be quantitatively described.

A representative example of such limitation is observed in bulk 2H-NbSe2, as mentioned above. The broad softening profile and the relatively weak thermal smearing effect in this material hinder the application of a discrete soft-mode analysis. While some theoretical studies have explored alternative mechanisms such as preformed excitonic states40, the prevailing consensus remains that electron–phonon coupling is the dominant driving force for CDW formation in 2H-NbSe23, albeit with features that fall outside the assumptions of our model.

In this context, the three-temperature model is broadly applicable to layered CDW materials where phonon softening plays a central role and the critical temperature lies in a moderate to high range. It can accommodate the coexistence of other electronic interactions, provided that the CDW transition involves well-defined soft-mode phonons. However, in materials with extremely low transition temperatures or where phonon softening cannot be unambiguously characterized, the model’s predictive capability is limited.

In summary, a three-temperature model used to describe the energy transfer process between electrons, soft-mode and non-soft-mode phonons has been developed. Mode-selective smearing induced soft-mode phonons can be quantitatively screened from phonon dispersions using different smearing parameters, based on the empirical relationship between ratios of soft-mode phonons standard coefficient and generalized effective atomic mass \({R}_{\gamma }={{R}_{{m}^{* }}}^{n}\). Corrected critical temperatures of different materials are consistent with the experimental results of the selected eight materials, except for some fluctuations near the smearing critical temperature. We believe that the proposed methodology would be able to predict CDW critical temperatures for materials with similar structures in a simple and efficient manner, if the parameter n were well fitted using one material as the reference, especially in the case of layered transition metal dichalcogenides. That is, to first determine the electronic critical temperature Te,c from the temperature-dependent phonon dispersions computed by Fermi smearing, and then to bring it into the three-temperature model to calculate the corrected critical temperature Te-p,c. We further believe that the three-temperature model could also be extended to other CDW systems or even to other phase transitions involving soft-mode phonons, such as quasi-one-dimensional CDWs and ferroelectric transitions, as well as pressure- or doping-induced CDW transitions, although this would require validation across a broad range of materials. Moreover, the proposed three-temperature model may provide a possible physical picture to understand the effect of smearing techniques not only on electrons but also on phonons, the role of soft-mode phonons in the process of mode-selective smearing and electron–phonon relaxation, and the relationship between phase transition and phonon softening from an atomic perspective.

Methods

First-principles calculation

Electron energy bands (see Supplementary Figs. S2c–f, S5a, and S7) and temperature-dependent phonon dispersions using smearing methods (see Figs. 2 and 5a–c and Supplementary Fig. S5b) of the eight materials were computed. Ab initio calculations were conducted using the Perdew–Burke–Ernzerhof generalized gradient approximation as implemented in the Vienna Ab-initio Simulation Package (VASP)41. Projector augmented wave42 pseudopotentials and an energy cutoff of 550 eV were employed. The lattice constants for TX2 were taken from bulk experimental values. Specifically, for 1T-TiSe2 with a space group (SG) of \({P}\overline{3}m1\), a = b = 3.540 Å, and c = 6.008 Å43; for 1T-TaS2 with a SG of \({P}\overline{3}m1\), a = b = 3.3672 Å and c = 5.9020 Å44; for 2H-NbSe2 with a SG of P63/mmc, a = b = 3.443 Å and c = 12.547 Å45; for 2H-TaSe2 with a SG of P63/mmc, a = b = 3.430 Å and c = 12.710 Å46; and for 2H-TaS2 with a SG of P63/mmc, a = b = 3.314 Å and c = 12.097 Å47. The experimental bulk lattice parameters with relaxed internal coordinates were used because the results show minimal sensitivity to the choice of exchange-correlation functional. Meanwhile, the optB86b-vdW functional48 was used to accurately describe van der Waals interactions. The 2D layers were modeled in periodic cells with a 15 Å vacuum in the out-of-plane direction to eliminate interaction between periodic images.

The Brillouin zone was sampled using a Γ-centered 18 × 18 × 11 (18 × 18 × 1) k-point mesh for bulk (monolayer) calculations. Energy and force convergence criteria were set to 10−8 eV and 0.001 eV/Å, respectively. The electronic structure calculations were performed with MedeA-VASP49,50. The Gaussian smearing method with a width of 0.05 eV was applied. The crystal structure and the electron charge density are visualized by the VESTA51.

Phonon dispersions were calculated using the PHONOPY code52, which utilizes the force constants obtained from the density functional perturbation theory53 as implemented in the VASP code. The calculations were performed with an energy cutoff of 550 eV and a convergence criterion of 10−8 eV for the total energy. Given the sensitivity of harmonic phonon frequencies to the chosen k-point sampling and electronic temperature Te = σ/kB, convergence tests were performed. For bulk (monolayer) 1T-TiSe2 and 1T-TaS2, a 4 × 4 × 3 (4 × 4 × 1) supercell and a Γ-centered 3 × 3 × 2 (8 × 8 × 1) k-point mesh were selected for phonon dispersion calculations. For bulk 2H-phase compounds, supercell sizes of 4 × 4 × 2, 6 × 6 × 1, and 6 × 6 × 1 were used for 2H-NbSe2, 2H-TaSe2, and 2H-TaS2, respectively. For monolayer 1H-NbSe2 and 1H-TaS2, a 6 × 6 × 1 supercell and a Γ-centered 4 × 4 × 1 k-point mesh were used. The electronic temperature Te for monolayer 1T-TiSe2 was simulated using various smearing methods, including the Fermi–Dirac, Gaussian, and MP functions for comparison, while the Fermi–Dirac function was employed for other cases. The computational parameters for phonon dispersions have been rigorously tested for convergence. The specific parameters and corresponding results are presented in Supplementary Fig. S1. As demonstrated by the convergence test, using 8 × 8 × 1 mesh and 9 × 9 × 1 mesh of calculation parameters, the phonon dispersion shows negligible variations.

Smearing methods

Smearing methods aim to improve the accuracy of k-point sampling and speed up the convergence of the Brillouin zone integration, when there is an abrupt change in the occupation function near the Fermi surface at low temperatures32. In smearing, an occupation function f(x) is defined as the integral of a broadening function \(\tilde{\delta }(x)\),

$$f(x)=\mathop{\int}\nolimits_{-\infty }^{x}\tilde{\delta} (\xi )d\xi ,$$
(20)

where \(x=\frac{\mu -\varepsilon }{\sigma }\). Commonly used smearing methods are Fermi–Dirac smearing, Gaussian smearing, and MP smearing. Corresponding broadening functions are listed as below:

$${\tilde{\delta }}_{Fermi}(x)=\frac{1}{2\cosh (x)+2},$$
(21)
$${\tilde{\delta }}_{Gaussian}(x)=\frac{1}{2\sqrt{\pi }}\exp (-{x}^{2}),$$
(22)
$${\tilde{\delta }}_{MP,N}(x)=\mathop{\sum }\limits_{n=0}^{N}\frac{{(-1)}^{n}}{n!{4}^{n}\sqrt{\pi }}{H}_{2n}\exp (-{x}^{2}).$$
(23)

Particularly, the occupation function of Fermi smearing \({f}_{Fermi}(x)=\mathop{\int}\nolimits_{-\infty }^{x}\tilde{{\delta }}_{Fermi}(\xi )d\xi ={[\exp (-x)+1]}^{-1}\) is the Fermi–Dirac distribution.