Introduction

The development of phononic devices1,2,3—which instead of charge carriers use phonons, the quanta of lattice vibrations, to encode information—is subordinated to finding materials with fast, reversible, and dynamical modulation of their thermal transport properties. Active control of the thermal conductivity, κ, also plays an important role in thermal management applications4 (e.g., to regulate batteries' self-generated heat5) and thermoelectric energy harvesting6. Nonetheless, dynamical κ modulation poses an extraordinary practical challenge: phonons are quasi-particles that neither have mass nor bare charge, thus controlling their propagation by means of external fields is far from straightforward. In this context, both experiments7,8,9,10 and theory11,12,13,14,15 have pointed to a promising route based on the writing/erasing of domain walls in ferroelectrics. More generally, external electric and magnetic fields, separately16,17,18,19,20,21,22 or jointly23, can be used to manipulate the lattice of polar/polarizable and magnetic materials, thus leading to large structural and thermal conductivity changes.

TiSe2 is a prototypical charge density wave (CDW) material that features a periodic modulation of the electron density below a critical temperature of Tc ≈ 200 K. Due to a coupling between the electronic and lattice degrees of freedom, the CDW originates a 2 × 2 commensurate lattice distortion (Fig. 2a). Interestingly, this CDW can be melted by impurity doping or light absorption24,25,26,27,28, leading to the stabilization of the undistorted 1 × 1 high-temperature phase. The physical mechanism driving this phase transition consists of the creation of a threshold density of mobile charges in the valence (holes) and/or conduction (electrons) bands. Experimentally, it has been shown that photoinduced CDW melting is an ultrafast and reversible process that occurs in the scale of picoseconds26. The thermodynamic and thermal transport changes associated with such a singular photoinduced CDW 2 × 2 ↔ 1 × 1 phase transition, however, remain poorly investigated to date.

The first proposals for a phononic transistor were entirely based on thermal concepts, where the heat flux between the source and drain was controlled by a thermal gate signal (i.e., the gate temperature)29. Yet, as mentioned above, alternative approaches in which the heat flux is modulated by an external field appear to be much more reliable and appropriate for practical implementations (e.g., faster commutation frequencies). In particular, the possibility of manipulating κ with light emerges as a very promising avenue since, despite having received very little attention thus far, is well suited for miniaturization and could simplify the design of logic devices (i.e., no need for electrical contacts).

In this paper, we investigate with theoretical first-principles methods the changes in thermal conductivity and thermodynamic properties of a TiSe2 monolayer (1L) upon photoinduced CDW melting at temperatures below Tc. A substantial κ reduction of up to 25% is predicted that we almost entirely ascribe to the changing anharmonic nature of a specific high-symmetry soft phonon mode driven by photoexcitation. In addition, a large light-induced phase transition entropy change is estimated which suggests the existence of sizable photocaloric effects in 1L-TiSe2. Notice that, although these effects should also be observed in few-layer and bulk TiSe2, which also feature a CDW instability30, light absorption can be achieved more efficiently in single layers and thin films and thus we focus on this system.

Methods

Lattice thermal conductivity

We performed density-functional theory (DFT) calculations with the VASP code31,32 using the projector-augmented wave (PAW) method33,34 with a plane-wave cutoff of 300 eV. We have used the local density approximation (LDA) for the exchange-correlation as parameterized by Perdew and Zunger35 to Ceperley and Alder data36. The different unit cells used as a basis for our whole set of calculations were optimized under strict criteria of convergence by relaxing the atomic positions until all forces were lower than 10−7 eV/Å with the lattice parameter fixed to the experimental value37,38,39 of 3.598 Å and sampling the Brillouin zone of the CDW 2 × 2 (undistorted 1 × 1) phase with an 18 × 18 × 1 (36 × 36 × 1) mesh of k-points.

The harmonic interatomic force constants (IFC), needed to compute phonon frequencies, eigenvectors, and group velocities, were obtained using the supercell method as implemented in phonopy40 using a 6 × 6 × 1 and 3 × 3 × 1 supercell for the undistorted 1 × 1 and CDW 2 × 2 phases, respectively. Similarly, third-order force constants, needed for the computation of three-phonon matrix elements were obtained with the method implemented in thirdorder.py41, using the same supercells with a cutoff, so that only interactions up to 0.73 nm (equivalent to 6th nearest neighbors of the 1 × 1 phase) were considered (convergence tests are shown in Fig. 1a).

Fig. 1: Convergence study of the thermal conductivity, κ, of the 2 × 2 CDW phase.
figure 1

a κ as a function of the cutoff beyond which interactions in the third-order force constants are neglected; the q-point mesh is 35 × 35 × 1; b κ as a function of different N × N × 1 mesh of q-points used to solve the BTE; the cutoff for third-order interactions is 0.73 nm.

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The IFCs were then used as input to solve the phonon Boltzmann transport equation (BTE) using almaBTE, in a 50 × 50 × 1 (25 × 25 × 1) Γ-centered q-mesh for the 1 × 1 (CDW) phase (see Fig. 1b). Within the relaxation time approximation (RTA), the lattice thermal conductivity reads

$${\kappa }_{\alpha \beta }=\sum _{\lambda }{C}_{\lambda }\frac{{v}_{\lambda \alpha }{v}_{\lambda \beta }}{| {{\boldsymbol{v}}}_{\lambda }| }{\Lambda }_{\lambda },$$
(1)

where the sum runs over all phonon modes, the index λ including both q-point and phonon band; Cλ is the contribution of mode λ to C(T), the volumetric heat capacity, v is the group velocity and Λλ is the mean free path. However, the RTA has some notorious shortcomings, which are particularly critical in two-dimensional systems, where momentum-conserving phonon–phonon processes are important. Therefore, here we adopt a full solution of the phonon BTE, beyond the RTA, that fully accounts for all scattering terms. The lattice thermal conductivity is obtained as

$${\kappa }_{\alpha \beta }=\sum _{\lambda }{C}_{\lambda }{v}_{\lambda \alpha }{F}_{\lambda \beta },$$
(2)

where Fλ is a generalized mean free path.

We used a broadening parameter fixed to 1 for energy conservation within a symmetric adaptive smearing approach42, which is a modification of the original scheme41 to properly account for microscopic reversibility and permutation symmetry:

$$\delta ({\omega }_{i}\pm {\omega }_{j}-{\omega }_{k})\approx \frac{1}{\sqrt{2\pi }{\sigma }_{ijk}}{e}^{\frac{-{({\omega }_{i}\pm {\omega }_{j}-{\omega }_{k})}^{2}}{2{\sigma }_{ijk}^{2}}}$$
(3)
$${\sigma }_{ijk}=a\sqrt{{\sigma }_{i}^{2}+{\sigma }_{j}^{2}+{\sigma }_{k}^{2}},$$
(4)
$${\sigma }_{i}=\frac{1}{\sqrt{12}}\left\Vert {\left\{{G}_{\mu \alpha }^{T}\cdot {N}_{\mu \mu }^{-1}\right\}}^{T}\cdot {({v}_{i})}_{\alpha }\right\Vert ,$$
(5)

where i, j, and k indicate phonon states, μ indicates a reciprocal-space lattice vector, α indicates a Cartesian axis, Gμα is the reciprocal lattice, Nμμ is a diagonal matrix with the q-grid size, a is the broadening parameter, and v is the phonon group velocity.

Light excitation is effectively accounted for through the use of the smearing parameter that controls how the partial occupancies are set for each orbital. We used a Gaussian smearing for VASP calculations, while we opted for Fermi–Dirac smearing in the electronic transport calculations described below, providing thus full consistency between the electronic temperature and smearing width, which is necessary for the computation of transport coefficients. A correspondence between the two smearing schemes has been obtained by mapping the evolution of the frequency of the soft mode at the M point (i.e., we considered two smearing widths to be equivalent if they yielded the same frequency for the soft mode). For the case of doping, harmonic IFCs used for the computation of phonon dispersions (top row of Fig. 2 in the main text) were obtained by adding a finite electron density ne to the self-consistency cycle (and thus to the conduction band) of the structural relaxation of the unit cell; in the 6 × 6 × 1 supercell calculations needed to calculate finite differences of the forces, the doping charge was appropriately rescaled like 62ne. For the case of photoexcitation, where ne = 0, harmonic and anharmonic IFCs were computed in the standard way without following any special prescription other than using the same smearing factor both in the unit cell and supercell calculations.

Fig. 2: Thermal conductivity of the low-temperature CDW ground-state.
figure 2

a Structure of the 2 × 2 CDW. The black and red arrows indicate the distortions, δTi and δSe, respectively (not in scale). b Phonon dispersion along the Γ–M–K–Γ path (Γ = 0, 0, 0; M = 1/2, 0, 0; K = 1/3, 1/3, 0). c, d Phonon velocities and lifetimes as a function of frequency. e Thermal conductivity as a function of temperature as obtained from the iterative solution of the linearized (continuous line) and the RTA (dashed line) phonon BTE.

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Electron–phonon scattering is not included in our lattice thermal conductivity calculations. Note, however, that neglecting such a source of possible phonon scattering is expected to be only relevant for the light-stabilized 1 × 1 undistorted phase, since this presents mobile charges in the conduction band. In fact, in a previous work by Dongre et al.43 it has been explicitly shown that phonon–electron scattering in archetypal insulator materials is only pertinent at high doping concentrations. As a consequence, the change of the thermal conductivity that we estimate turns out to be a lower bound of the actual value, since electron–phonon scattering would further decrease the κ of the light-stabilized phase.

Electronic thermal conductivity, Seebeck coefficient, and ZT

We used the QUANTUM ESPRESSO (QE) software44,45 with similar settings as those used in the lattice thermal conductivity calculations (e.g., PAW, energy cutoffs, the exchange-correlation functional,...), but including a self-consistently calculated Hubbard correction computed using the Hubbard parameters (HP) code46, as this shifts the transition range near the experimental value47. This last point is of key importance for electronic transport calculations, as it allows to use the same temperature for both the smearing in first-principles calculations and the occupations in the transport calculations. To further illustrate this point, if no Hubbard correction was considered, these two quantities would differ more than 1000 K47.

Harmonic IFCs and the first-derivative of the Kohn-Sham potential with respect to phonon perturbations were obtained using the density perturbation functional theory with Hubbard (DPFT + U), as implemented in the PHonon package included within QUANTUM ESPRESSO suit48, using a 12 × 12 × 1 Γ-centered q-mesh and a strict convergence limit of 1.0−16 Ry. Then, the computation of electron–phonon matrix elements and Hamiltonian in the real space was performed with the EPW package49, through the wannierization of the isolated manifold composed of 7th–15th band, using SCDM-k method50 as implemented in wannier9051 for the initial projection. We note that the electron–phonon matrix elements do not take into account the Hubbard contribution beyond phonon frequencies and the unperturbed electronic density; nonetheless, the introduction of such effects has been shown to introduce small shifts into the electron–phonon matrix elements52.

Following this procedure, the electronic thermal conductivity (κel), the diffusive Sebeeck coefficient (Sel), and the electrical conductivity (σ) were obtained by iteratively solving the electronic BTE with elphbolt53, until the relative change in all transport coefficients was less than 1.0 × 10−4.

These quantities together with the lattice conductivity at a given T allow for the computation of the thermoelectric figure of merit:

$$ZT=\frac{\sigma {S}_{el}^{2}}{{\kappa }_{ph}+{\kappa }_{el}}.$$
(6)

Notice that, though drag effects were neglected (i.e., the electron and phonon BTEs beyond RTA are solved independently), all ab initio quantities were calculated at that specific electronic temperature, so that we did not merely account for thermal effects on the occupation factors. In this respect, the determination of the electronic temperature for the Gaussian smearing calculation has been performed by matching their phonon frequencies—with a preponderance of the M-point—with the ones computed using a Fermi–Dirac smearing.

Results

Phonon transport in the CDW ground state

Below the critical temperature Tc, which experimentally varies between ≈200 K54 and 232 K55, 1L-TiSe2 exhibits a commensurate CDW 2 × 2 phase. While in the undistorted high-temperature 1 × 1 phase there is only one type of octahedra (with six Ti–Se bonds of identical length), in the 2 × 2 distorted ground state there are two different types of octahedra per unit cell, one of type I and three of type II. Type I octahedra exhibit all six Ti–Se bonds of identical length and are very similar to those found in the undistorted 1 × 1 phase; type II octahedra, which are those more severely distorted, exhibit two short, two long, and two intermediate Ti–Se bonds, the latter being very similar to those found in type I octahedra. The Ti and Se displacements occur mostly parallel to the layer and, according to an X-ray study56, the ratio between the Ti and Se displacements amounts to 3.3 (δTi = 0.082 Å and δSe = 0.025 Å at T = 77 K), which is very similar to the value of 3.0 found in bulk TiSe257 (note that in bulk the CDW also leads to doubling of the unit cell along the interlayer direction). The influence of these structural parameters on the electronic properties of 1L-TiSe2 has been analyzed in detail in a previous work58.

Following Calandra and Mauri37, we kept the lattice parameter fixed to the experimental value of 3.5398 Å and optimized the atomic coordinates. It has been shown38 that this choice yields phonons in excellent agreement with the experiments59. We obtained a displacement of the Ti atoms of 0.075 Å and a corrugation of 0.008 Å, in agreement with previous results39,55,58,60. The phonon dispersion shown in Fig. 2b is well-behaved and does not present imaginary frequencies, thus corroborating the expected dynamical stability of the distorted 2 × 2 ground-state phase. Intriguingly, the largest calculated phonon velocities correspond to optical modes, not to acoustic modes, in the frequency range 6 ≤ ω ≤ 8 THz (Fig. 2c). However these optical modes present very strong anharmonic scattering, with lifetimes that are 2–3 orders of magnitude smaller than those of low-frequency phonons (ω < 2 THz), and consequently their contribution to κ turns out to be negligible (Fig. 2d).

Our results for the thermal conductivity of the CDW 2 × 2 phase in the temperature range 100 ≤ T ≤ 200 K are shown in Fig. 2e. The CDW becomes unstable at temperatures above 200 K, while below 100 K is difficult to guarantee robust convergence of the computed κ values with respect to the number of q-points used to solve the BTE, due to the importance of very long wavelength phonons. For the calculation of the thermal conductivity of 2D materials61,62,63,64, one usually needs to adopt a particular layer thickness. Here, we assumed it to be equal to 6 Å, that is, the interlayer separation in bulk TiSe265. Although there are more sophisticated procedures for selecting this parameter (e.g., based on the decay of the electron density away from the atomic planes), here we are mainly concerned with the variation of κ upon melting of the CDW (i.e., in going from the 2 × 2 CDW to the 1 × 1 undistorted phase), hence any reasonable choice of the layer thickness may be expected to provide equivalent results. Under these assumptions, we obtained a κ value of 10 W m−1 K−1 at 200 K. It is worth noting that the use of the RTA, which is notoriously unreliable for 2D materials where hydrodynamic phonon transport plays an important role66,67,68, in fact, provides a κ underestimation of approximately 10%. This result stems from the fact that the RTA treats as resistive even those phonon–phonon processes that simply redistribute momentum, and is common to a broad class of 2D materials.

Light-induced CDW melting

At temperatures below Tc, the undistorted 1 × 1 phase is dynamically unstable. In our zero-temperature calculations, such a vibrational instability manifests as an imaginary phonon mode at the high-symmetry reciprocal point M (Fig. 3). This zone-boundary instability has been widely reported in the literature39,58,60,69 and directly hints at the 2 × 2 periodicity of the CDW phase. Nonetheless, this vibrational instability can be lifted either by electron/hole doping or photoexcitation, which we effectively accounted for by means of the smearing of the Fermi–Dirac distribution that determines the partial occupancies of each orbital, as discussed in the Methods section. In the first case, excess charge is directly injected from the electronic states of extrinsic impurities into the conduction/valence band; in the second case, excitons are photogenerated and electrons are promoted from the valence to the conduction band. Both routes leading to suppression of the CDW instability have been already reported58,69,70,71. For instance, experimental evidence for photoinduced CDW melting has been provided by Möhr-Vorobeva and coworkers26, who also have reported that the CDW 2 × 2 → 1 × 1 phase transition takes place during a very short time interval of 250 fs (recovery of the CDW instability upon removal of the light source takes a bit longer, 10 ps–100 ns). Figure 3 illustrates both CDW suppression mechanisms as obtained from first-principles calculations. Therein, we plot the phonon dispersion curves obtained for different excess charge carrier densities in the conduction band, ne. Note that in the photoexcitation case ne is always compensated by an equal density of excess holes, nh, in the valence band (in contrast to the extrinsic impurities case, in which only ne or nh are present). It is worth mentioning that biaxial strain (or pressure in the case of bulk TiSe272) may also drive the stabilization of the undistorted 1 × 1 phase at low temperatures, as it has been experimentally73,74 and theoretically70,75 demonstrated (we will not explicitly address this latter case here).

Fig. 3: Lifting of the dynamical instability of the 1 × 1 phase.
figure 3

Phonon dispersion of the 1 × 1 phase for different concentrations, ne, of charge carriers in the conduction band provided by impurity doping (top row) or photoexcitation (bottom row). The phonon dispersion of the undoped 1 × 1, exhibiting the characteristic instability at the M point, is shown on the left-hand side.

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Two important observations can be made by looking at Fig. 3. First, doping and photoexcitation may provide almost identical phonon dispersion curves although for quite different ne values (i.e., in the photoexcitation case a lower ne is needed for CDW melting since, as mentioned above, ne is always mirrored by an equal density of holes, nh, which also contributes to the stabilization of the undistorted 1 × 1 phase58). Second, and most importantly for κ tuning, photoexcitation acts almost exclusively on one single phonon mode, the soft mode at the high-symmetry reciprocal point M that drives the CDW instability, leaving the rest of the phonon dispersion essentially unaltered. It is worth noting that, right upon stabilization of the undistorted 1 × 1 phase this soft phonon mode acquires a “V” shape with an almost linear dispersion around the M point, which is reminiscent of Γ acoustic modes. Indeed, at low charge injections, the soft mode keeps a close resemblance with the unstable mode in dark conditions that exhibit a minimum at the M point. As ne is steadily increased, the frequency of the M phonon mode also increases, eventually merging with the mid-frequency phonon bands.

In order to check the effect of temperature on the phonon dispersion, we have computed finite temperature phonons at 200 K with a normal-mode-decomposition technique76, for the 2 × 2 CDW distorted phase. The comparison with the zero-kelvin phonon dispersion (see Supplementary Fig. 4) suggests that this is a reasonably good approximation. The only noticeable difference is a softening of the highest energy acoustic mode, which deserves further investigation (differences in the optical modes at ω > 6 THz are not playing a role in heat transport; see below the discussion of Fig. 5). Notice also that excitonic effects are not accounted for in our calculations. It has been shown that including them has the main effect of reducing optical transition energies77, which we have not addressed in the present manuscript.

Thermal conductivity of the 1 × 1 phase

Our results for the thermal conductivity of the photo-stabilized 1 × 1 phase are shown in Fig. 4, where we plot the relative difference, Δκ, with respect to the CDW 2 × 2 ground state. As it can be appreciated there, for high enough ne values the thermal conductivity of the 1 × 1 phase practically matches that of the CDW phase, while for more moderate ne’s a considerable reduction of up to 25 % is evidenced.

Fig. 4: Thermal conductivity of the CDW-melted 1 × 1 phase.
figure 4

Reduction of the thermal conductivity, Δκ, as a function of temperature upon the light-induced melting of the CDW for different concentrations of the photoexcited charge. Inset: phonon dispersion for the values of ne of the main panel.

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These results in principle may seem counter-intuitive, since, as a general rule, materials with smaller/simpler primitive cells—like that of the 1 × 1 phase as compared to the CDW ground state—tend to exhibit larger κ values. The reason for this argument is that reduced primitive cells imply smaller numbers of phonon modes hence the occurrence of energy and momentum-conserving phonon–phonon processes, which can reduce κ, is less likely. In the present case, these qualitative arguments, however, only apply to high ne values for which the main effect of symmetry breaking is to lift a few phonon degeneracies. Figure 5, for instance, shows the phonon dispersion curves obtained along the Γ–M path for the CDW ground state and a strongly doped 1 × 1 phase; to allow for a direct comparison, the bands of the latter phase were folded, as if they were calculated in a 2 × 2 supercell. The effects of symmetry breaking are most significant for the optical phonon modes with frequencies 6–7 THz, which split into four branches in the CDW phase. However, phonon modes in that frequency region do not appreciably contribute to κ, as it is shown by the cumulative thermal conductivity plot in the same figure (see also Fig. 2 and related discussion). Differences are also appreciated at lower frequencies, although most of them concern dispersionless optical modes that do not carry heat due to their very low phonon velocities. Heat-carrying acoustic modes, on the other hand, hardly exhibit any difference between the two phases.

Fig. 5: High photoexcitation limit.
figure 5

(Left) Comparison between the phonon dispersion of the CDW (black) and the 1 × 1 phase (red) along the Γ–M path for ne = 0.074640; the latter has been folded to allow a direct comparison. (Right) Cumulative thermal conductivity as a function of phonon frequency of the CDW phase.

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By looking at the evolution of phonon dispersion curves in Fig. 3, it is clear that the qualitative explanation given above does not apply to the results obtained for low ne values. At these conditions, the differences between the phonon bands of the two phases are significant, the 1 × 1 soft phonon mode with a pronounced minimum around the zone-boundary point M playing a major role. At first glance, however, the Δκ results obtained in this case also seem surprising: the quasilinear dispersion of the 1 × 1 soft mode around the M point (e.g., case ne = 0.075 in Fig. 3) indicates large phonon velocities, thus suggesting an increase in the thermal conductivity, while the explicitly computed κ values may be ≈25 % lower. Nevertheless, we found that phonon–phonon scattering events involving the M soft phonon mode turn out to be especially resistive, as shown in Supplementary Figs. 1 and 2, which account for the resistive character of the collision processes. Moreover, those maps also reveal that the acoustic phonon branches in the 1 × 1 phase follow the same trend. Thus, the relative κ decrease predicted here for the undistorted phase mainly originated from a surge in the number of resistive phonon–phonon collisions ascribed to the M soft mode and heat-carrying acoustic modes.

The usual classification of phonon–phonon processes into “Umklapp” and “Normal” has been shown to be not rigorous since it relies on the choice of the primitive cell78. Here, nonetheless, it serves as a simple, although qualitative, interpretation of the observed behavior. The low-frequency part of the soft phonon mode lies very close to the Brillouin zone boundary (i.e., the M point) hence practically all the allowed phonon–phonon processes involving it will have a quasimomentum lying outside the first Brillouin zone, which is the signature of Umklapp processes. These processes destroy phonon momentum and are purely resistive, in accordance with the values shown in Supplementary Fig. 2.

As a final remark, it is important to stress that the κ differences between the CDW 2 × 2 and undistorted 1 × 1 phases can be almost entirely traced back to one single phonon mode. The rest of the phonon dispersion curves hardly undergo any change under varying ne. This can be clearly appreciated in the inset of Fig. 4, where the undistorted 1 × 1 phonon bands corresponding to thermal conductivities represented in the same figure as a function of ne appear virtually indistinguishable. A zoomed-in view of the phonon branches at wave vectors around the M point is necessary to reveal definite differences.

Electronic thermal conductivity

The absence vs occurrence of a small bandgap for undistorted 1 × 1 1L-TiSe2 is unclear, due to the difficulty in preparing non-doped samples. However, experimental results clearly show a drastic reconstruction of the electronic structure and a semiconducting character when the 2 × 2 CDW occurs54. These considerations can be important for the κ tuning discussed because a metallic 1 × 1 phase would also feature a non-negligible contribution to the total thermal conductivity, which could partially compensate for the effect reported above (the 1 × 1 phase has a lower phononic conductivity than the CDW phase).

In view of this possibility, we explicitly calculated the electronic contribution to the thermal conductivity for the undistorted 1 × 1 phase, κel, along with some related quantities like the diffusive Sebeeck coefficient, Sel, and electrical conductivity, σ. We find it worth mentioning that, due to the limitations of LDA in providing accurate band structure for highly correlated systems and the sensitivity of the electronic properties to this, we employed LDA + U to better match experimental results. The on-site Coulomb interaction (U) was determined self-consistently for different smearing values, something crucial for an accurate description of the CDW transition39,47. According to the results shown in Table 1, thermal transport in the 1 × 1 phase is almost entirely governed by the atomic lattice since the electronic thermal conductivity can be neglected to all practical effects. Consequently, the large photoinduced κ changes driven by the melting of the CDW are predominantly of phononic nature. Furthermore, the results in Table 1 also show that although the 1 × 1 phase cannot be regarded as a superb thermoelectric material (i.e., the corresponding thermoelectric figures of merit, ZT, approximately amount to 0.03) is still comparable in performance to some other well-known semiconductors (e.g., doped Si, which as a reference displays a ZT in the range of 0.004–0.011 at around 200 K79), thus offering room for further improvements.

Table 1 Thermoelectric properties
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Envisaged applications: solid-state cooling

Besides the realization of phononic devices1,2,3, the photoinduced 2 × 2 CDW → 1 × 1 phase transition in 1L-TiSe2 also holds great promise for developing solid-state cooling applications based on thermoelectricity80,81 (see above) and novel caloric effects82,83, as we explain next.

Under external field (e.g., pressure and electric bias) variations, caloric materials experience large adiabatic temperature variations (∣ΔT∣ ~ 1–10 K) as a result of phase transformations entailing large isothermal entropy changes (∣ΔS∣ ~ 10–100 J K−1 kg−1). Such caloric effects may be exploited for engineering solid-state refrigeration cycles82,83. The large photoinduced lattice κ variation unraveled here for TiSe2 hints also at the existence of large isothermal entropy changes driven by light absorption (i.e., photocaloric effects). In fact, our first-principles calculations render sizeable ∣ΔS∣ values of ≈6 J K−1 kg−1 for photoexcited charge densities of around 0.074 e Å−3 at T = 200 K, 2 × 2 CDW being the phase presenting the lowest entropy.

A possible four-step refrigeration cycle based on the photocaloric effect disclosed here for TiSe2 may be envisaged (see Fig. 6). First, starting from the low-entropy 2 × 2 CDW phase at T0 < Tc, the system will be adiabatically irradiated with light until stabilizing the high-entropy 1 × 1 phase. As a consequence, the temperature of the TiSe2 sample will decrease, T1 < T0 (i.e., inverse photocaloric effect, ΔT < 0). Second, by keeping the light conditions on, the sample will be put into contact with the targeted body to be refrigerated so that heat, ∣Q∣ = T∣ΔS∣, will be transferred to TiSe2 until restoring the initial temperature conditions, T2 = T0. Third, the light irradiation will be adiabatically removed thus stabilizing again the low-entropy 2 × 2 CDW phase. Consequently, the temperature of the TiSe2 sample will increase, T3 > T0. Fourth, the sample will be put into contact with a heat sink so that heat will be removed from TiSe2 until restoring the initial temperature conditions, T4 = T0, thus completing a cooling cycle. An approximate estimation of the efficiency of the four-step refrigeration cycle just described, which considers ∣Q∣, the amount of heat associated with the photocaloric effect, and the minimum electronic work that is necessary to trigger the CDW melting in TiSe2, amounts to 34%.

Fig. 6: Photocaloric cycle.
figure 6

Sketch of the four-step refrigeration cycle based on the photocaloric effect.

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Such a possible solid-state cooling cycle presents a clear drawback: the operation conditions for TiSe2 should be T0 < 200 K, which is well below room temperature. (This practical limitation, however, may be overcome by finding analogous CDW materials presenting higher critical temperatures.) On the other hand, the described refrigeration cycle is very promising in several aspects, namely, (1) the temperature span over which it could be operated is unusually ample (~100 K) since the underlying photoinduced phase transition is not restricted from below Tc, (2) due to the two-dimensional nature of TiSe2, light absorption may be achieved very efficiently, and (3) the proposed refrigeration cycle is well suited for miniaturization. Noteworthy, an analogous cycle with a giant room-temperature photocaloric effect has been recently reported in a ferroelectric perovskite84.

Discussion

We have reported first-principles calculations hinting at a dynamical and reversible modulation of the thermal conductivity of 1L-TiSe2 driven by photoinduced melting of its CDW. Such changes occur in an ultrafast timescale of picoseconds and can be also obtained by field-controlled activation of impurity doping. The lower thermal conductivity of the undistorted 1 × 1 phase stems from the resistive character of phonon–phonon processed involving a high-symmetry soft mode, which otherwise is responsible for the stabilization of the distorted CDW 2 × 2 phase. These findings should be general to few-layer and bulk TiSe2, featuring an analogous lattice distortion below a similar critical temperature, and to other CDW materials30, and thus open new avenues in the design of thermal devices in which information is encoded through phonons and that require of materials with fast and dynamical thermal responsiveness. The existence of sizeable photocaloric effects in 1L-TiSe2 is also proposed, which may find applications in solid-state nanocooling.