The critical role of hot carrier cooling in optically excited structural transitions

The hot carrier cooling occurs in most photoexcitation-induced phase transitions (PIPTs), but its role has often been neglected in many theoretical simulations as well as in proposed mechanisms. Here, by including the previously ignored hot carrier cooling in real-time time-dependent density functional theory (rt-TDDFT) simulations, we investigated the role of hot carrier cooling in PIPTs. Taking IrTe2 as an example, we reveal that the cooling of hot electrons from the higher energy levels of spatially extended states to the lower energy levels of the localized Ir-Ir dimer antibonding states strengthens remarkably the atomic driving forces and enhances atomic kinetic energy. These two factors combine to dissolute the Ir-Ir dimers on a timescale near the limit of atomic motions, thus initiating a deterministic kinetic phase transition. We further demonstrate that the subsequent cooling induces nonradiative recombination of photoexcited electrons and holes, leading to the ultrafast recovery of the Ir-Ir dimers observed experimentally. These findings provide a complete picture of the atomic dynamics in optically excited structural phase transitions.


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The photoinduced phase transitions (PIPTs) have become an appealing approach exploring ultrafast change and manipulation of material properties owing to the recent advances in ultrafast time-resolved diffraction techniques, combining ultrafast temporal manipulation with atomic-scale spatial resolution.  The optical excitation induces a nonequilibrium occupation of excited electronic states, which could lead to periodic lattice distortions (PLDs), expose the transient metastable states 23,24 , or yield a controllable phase transition to the desired phase for practical applications 5,9,10,21 . By means of photoexcitation, ultrafast phase transitions have been realized in quasi-one-dimensional (1D) 5,6,10,16,17 , two-dimensional (2D) 3,7,15,20,22 and three-dimensional (3D) 9,12,18,19,21 systems. One commonly used picture to explain the phase transition is the following: the photoexcited occupation of higher electronic states modifies the energy landscape substantially so that the original metastable phase becomes now a lower energy stable phase than the original ground state phase, causing the phase transition dynamically along the potential energy surfaces (PESs). 5,7,9,25 An alternative, yet related, the picture is that the transient change in the PESs results in a non-thermal excitation of soft phonon modes, which leads to a critically damped nuclear motion following these soft phonon modes to the end phase of the PIPT. 5,25 Our previous work 26 has also pointed out that the atomic forces for driving the PIPT in IrTe2 arising from occupation of the Ir-Ir dimer antibonding (bonding) states by optically excited electrons (holes). 26 All these proposed explanations consider the photoexcited carriers as the cause for the change of PESs or the generation of additional atomic driving forces but disregard completely the phenomena of their relaxation to lower energy levels that occurred within the first hundred femtoseconds after photoexcitation.
Evidence has accumulated that hot carrier cooling may play an essential role in photoinduced ultrafast structural phase transitions 7,8 . For instance, Ideta et al. attributed the observed ultrafast recovery of Ir-Ir dimers following their dissolution to carrier recombination. 7 Monney et al. postulated that partial PIPT is driven by a transient increase of the lattice temperature following the hot carrier cooling in picoseconds. 8 We have also demonstrated that one can control the structural phase transitions by selectively exciting photocarriers into designated excited electronic states. 26 The hot carrier cooling will undoubtedly alter the carrier's occupation to the excited states, thus changing the atomic driving forces. Besides, there are many possible effects in a PIPT, including electronelectron and electron-phonon interactions and thermal fluctuations. The lack of real-space atomic snapshots and the inability to turn on and off physical effects (e.g., electron-electron 3 / 20 and electron-phonon interactions) experimentally renders it challenging to disentangle these interweaving physical effects relying on experimental measurements alone. Here, by including the previously ignored hot carrier cooling effect, we have advanced the real-time time-dependent density functional theory (rt-TDDFT) simulations to study the dynamical processes of PIPT.
In contrast, most current rt-TDDFT simulations can only describe the excited systems' immediate dynamics following the photoexcitation since they cannot describe the hot carrier cooling effect. Furthermore, in many simulations, optically excited electrons are mimicked by manually taking electrons from the top of the valence band and placed at the bottom of the conduction band, or thermally distributed using a very high temperature. 5,9,10,25,27,28 Such simulations may provide a qualitative picture for PIPT, but it is unfeasible to reproduce the transition times behavior observed in the experiments accurately. An improvement has been made in some recent rt-TDDFT simulations using a Gaussianenvelope laser pulse to represent the actual laser light. 24,26,29 But the use of Ehrenfest dynamics in the rt-TDDFT is unlikely to describe the carrier cooling correctly since it lacks the detailed balance. As a matter of fact, the Ehrenfest dynamics tend to overheat the electronic subsystem. In this work, to include the hot carrier cooling effect, we have improved our rt-TDDFT algorithm by including a Boltzmann factor, 30 which can restore the detailed balance between various electronic state transitions and hence treat the carrier cooling properly. Figure 1 shows that after including the hot carrier cooling effect, the TDDFT simulation can precisely reproduce the experimentally measured femtosecond electron diffraction (FED) curve responsible for the complex phase transitions of IrTe2 (including dissolution and recovery of Ir-Ir dimers) following photoexcitation. More specifically, the inclusion of the hot carrier cooling process in the state-of-the-art rt-TDDFT simulation significantly accelerates the phase transition within 300 fs compared to the usual rt-TDDFT simulations, which display no transition throughout the 1.2 ps simulation time. We will also explicitly illustrate that Ir-Ir dimers' ultrafast recovery following their dissolutions is caused by electrons' continuous cooling passing through the Fermi level towards nonradiative recombination with photoexcited holes. These results demonstrate that hot carrier cooling plays a vital role in the photoexcitation-induced structural phase transition. in the TDDFT simulation without (with) the carrier cooling.

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To reveal the mechanism underlying the hot carrier cooling, we first examine the situation neglecting the hot carrier cooling process. More specifically, a straight forward rt-TDDFT simulation is carried out for the IrTe2 system irradiated by a femtosecond laser pulse (central wavelength 400 nm, pulse duration 120 fs) with its amplitude tuned so that 3% of valence electrons is optically excited from the valence band to the conduction band (same as in experiments 7 ). Figure. 2A shows that immediately following the photoexcitation, only 40% of the empty Ir-Ir dimers' antibonding states are filled, and the rest of the photoexcited electrons occupy the higher energy states. In our previous work, we have demonstrated that the excited electrons occupying such higher energy states tend to suppress the LT-to-HT phase transition. 26 We then proceed with the atomic dynamics by performing the rt-TDDFT simulation without including the Boltzmann factor as usual. As expected, no significant energy transfer occurs from photoexcited electrons to the lattice, and the lattice temperature remains around 200 K throughout the simulation. The top panel in Fig. 1B shows that, within the 1.2 ps simulation time, it is unlikely that the Ir-Ir dimers will undergo dissolution to achieve the LT-to-HT phase transition, which is quantified by the increase of Ir-Ir dimer bond length from 3.1 to 3.9 Å. One reason is that the photoelectrons fill only 40% of the Ir-Ir dimer's antibonding states, which is impossible to produce a strong enough atomic force to drive the phase transition. In reality, hot carriers tend to relax to lower energy electronic states and give the released energy to the lattice through electron-phonon interaction. [1][2][3][11][12][13][14][15] This transfer of the energy also heats the lattice subsystem. To explore the hot carrier cooling effect, we carry out the rt-TDDFT simulation again but by adding a particular Boltzmann factor in the algorithm 30 . Figure 2B shows the dynamic evolution of excited electrons and holes following photoexcitation. It is necessary to evaluate the dynamic filling of the Ir-Ir dimers' antibonding states by electrons considering their occupation responsible for the atomic force driving the structural phase transition 26 . To quantify it, here, we define as the integration of the time-dependent (due to hot carrier cooling) electronic occupation of these states (located about 0.3-1.0 eV above the Fermi level as indicated in Fig. 2): . .
. Figure 3A shows the development of : it grows at first 120 fs from zero to a saturation value (120 fs) as a result of pulse laser photoexcitation, and then declines slightly from 120-150 fs, which indicates the emerging of hot carrier cooling process. Such decline is due to a net loss of excited electrons in the Ir-Ir dimers.
The excited electrons belonging to them relax to lower energy states of other atoms but get fewer electrons from higher energy states. Figure 3B shows that, during the period, the atomic kinetic energy of the Ir-Ir dimers stays at a low level, indicating the electronelectron interaction rather than phonon-assisted process dominates the carrier cooling.
During this period, the Ir-Ir dimers' bond length exhibits an invisible change (Fig. 1B).
The situation changes dramatically after 150 fs. Figure  To verify the effect of the increase in the atomic kinetic energy of Ir dimers unambiguously, we re-do our simulation in the NVT ensemble during carrier cooling. In contrast to the above-adopted NVE ensemble, which considers the energy transfer from hot carrier cooling to the kinetic energy in the transition degree of freedom, 30 in the NVT the lattice kinetic energy is kept constant for a given initial temperature to mimic a situation for heat dissipates quickly. Figure S2 shows that in this case, the increase in the occupation of the Ir-Ir dimers' antibonding states strengthens the atomic driving forces following hot carrier cooling. However, the Ir-Ir dimers are unable to be dissociated due exclusively to the absence in the enhancement of the Ir-Ir dimer kinetic energy. Considering the lattice can only oscillate half of its vibrational period during hot carrier cooling, as discussed above, it is difficult to claim that its kinetic energy will be dissipated out as heat during that time. Thus, we believe the NVE simulation is more closed to reality. We subsequently conclude that the hot carrier cooling inducing the increase of Ir dimer kinetic energy plays a vital role in the structural phase transition.
Furthermore, the enhanced atomic forces drive the Ir-Ir dimers' dissolution to have a deterministic and coherent manner without exhibiting a significant random fluctuation. After reaching the minimum at around 750 fs, Fig. 1a shows that the diffraction intensity change begins to rise, which was postulated as the recovery of the Ir-Ir dimerization. 7 Here, we confirm this postulation theoretically and reveal the underlying mechanism. The difference from semiconductors, IrTe2 lacks a bandgap. Thus, the cooling of the optically excited electrons is unlikely to end at the bottom of the conduction band due to the phonon bottleneck as in semiconductors. Instead, excited electrons cool down continuously, passing the Fermi energy to finally fill the holes in the valence bands, making nonradiative recombination of electrons and holes without photons' emission. Figure 3B shows that the number of excited electrons starts to drop rapidly at 210 fs, indicating the electron-hole nonradiative recombination that releases the atomic driving forces exerting on the Ir-Ir dimers. At this moment, these Ir atoms have been accelerated to high speeds.

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These residual speeds drive them towards the nearest neighbor of non-dimerized Ir atoms to form new dimers (Fig. 4). Interestingly, these new dimers are formed even though their lattice temperature (T ~ 900 K) is well above the critical temperature TS = 280 K of thermalinduced phase transition 36 . It shows, once again, the phase transition is a coherent kinetic process, not a thermodynamic random process. Although the atomic kinetic energy is important (as demonstrated in the NVE simulation) in a photoinduced phase transition, its role is unlikely as in the thermodynamic phase transition but a kinetic movement to overcome barriers towards breaking the old dimer and forming the new dimers.
Accompanying the new Ir-Ir dimerization, the vanished antibonding states of Ir-Ir dimers above the Fermi level re-appear in the PDOS (see Fig. S3). Therefore, we have developed a microscopic explanation for the observed ultrafast recovery of PLD in experiments 7 . We also record the structural phase change evolution dynamics following photoexcitation in a movie (given in SI). In short, we reproduced the experimentally measured change of the diffraction intensity following the photoexcitation, in both magnitude and time scale. All this is only possible after we take into account the carrier cooling effect.
Our excellent agreement with the experiment also supports a fast sub-picosecond energy transfer from electron to phonon. It contrasts with a common perception that the electron to phonon energy transfers is a slow process in several ps, as judged by the observed electron-phonon equilibration time (several ps). 39,40 We believe this long electron-phonon equilibration time is mainly due to the equilibration process of lattice thermal motions following the initial coherent motions with atomic kinetic energy obtained from the carrier cooling. After the lattice vibration reaches equilibrium, the system will finally transfer from the new PLD structure to the HT phase due to its high lattice temperature (T ~ 900 K, which is well above the critical temperature TS = 280 K). We indeed notice that a much slower phase transition following photoexcitation has also been reported experimentally in IrTe2. 8 However, this should be distinguished from the subpicosecond diffraction measurement shown in Fig. 1A. To examine the temperatureinduced structural phase transition, we utilize a Born-Oppenheimer MD (BOMD) to simulate the lattice dynamics of the system in the ground state at lattice temperature ≈ 1000 K but without photoexcitation (Fig. S4). We indeed find that the Ir-Ir dimers undergo a dissolution at about 6.5 ps, a time scale nearly one order of magnitude slower than the PIPT. In this case, high lattice temperature is the only aspect responsible for this structural phase transition. The widespread distribution of photoexcited carriers will not exert a sufficiently large atomic force to dissociate the Ir-Ir dimers. On the other hand, the hot carrier cooling will have two major effects, which help the Ir-Ir dimer dissolution: first, there will be more occupation of the Ir-Ir dimer antibonding state, which enhances the atomic force to dissociate the dimer; second, the atoms of the Ir-Ir dimers will gain considerable coherent kinetic energies, which can help them to overcome any barrier during the dimer breaking process. In a combination of these two factors, the hot carrier cooling yields a phase transition curve in excellent agreement with the experimentally observed time-resolved diffraction data. We further show that the transition is deterministic and coherent. The coherent kinetic energy induces the formation of the new Ir-Ir dimers (at different locations from the initial ones), contributing to the experimentally observed sub-picosecond 11 / 20 recovery of the LT phase. On the other hand, for a thermal equilibrium system to induce phase transition due to its high-temperature effect, it can take about 6.5 ps, ten times longer than the fast coherent phase transition caused by hot carrier cooling. We believe our current understanding presents a new insight into photoexcitation-induced ultrafast phase transitions. It is likely also applicable to other ultrafast phase transitions for systems like vanadium dimers in VO2 9,18 and CDW material 1T-TaS2 1 , 1T-TiSe2 2,14 , and 1T-LaTe3. 13

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To mimic the photoexcitation, we apply an external electric field to simulate a laser pulse with a Gaussian shape in our rt-TDDFT, where E0 = 0.2 V/Å, t0 = 60 fs, 2 = 25 fs is the pulse width, and  = 3.1 eV is the photon energy. 7

Boltzmann Formula in rt-TDDFT:
To depict hot carrier cooling, we utilize the Boltzmann formula in rt-TDDFT. 30 Boltzmann formula will depend on adiabatic state ( ) in eq. 1. After Boltzmann correlation ( / , , here, , is a decoherence time of 20 fs ), a change  , ( ) is introduced in the coefficient , ( ), thus, the wave function is updated Compared with total energy (Etot1) calculated in eq. 1, the total energy (Etot2) is indeed changed in eq. 4. Based on the change of energy, we can obtain the increased kinetic energy which embodies the el-ph coupling effect. This energy will be added to the transition degree of freedom in the NVE ensemble. For a more detailed description of the Boltzmann factor formalism, we refer to it in Ref. 30. In the NVT ensemble with carrier cooling, the total kinetic energy of all the atoms is kept the same by a simple rescaling.

Section 2. Dynamics of thermally induced structural phase transitions
The hot carrier cooling following the photoexcitation heats the lattice subsystem to above the critical temperature for temperature-induced structural phase transition in IrTe2.
It is interesting to investigate the thermal effect on photoinduced structural phase transitions. To study the temperature-induced structural phase transitions, we utilize Born-Oppenheimer MD (BOMD) to simulate the dynamics of phase transition at lattice temperature ≈ 1000 K as shown in Fig. S7(A). One can see in Fig. S7(B) that the LTto-HT phase transition occurs at 6.5 ps as characterized by the transferring of the 1/5 periodic charge-modulation structure (LT phase) to an undistorted lattice (HT phase). Note that thermally driven phase transition within the several-picosecond timescale belongs to disordering dynamics. This is nearly 10 times slower than the coherent dissociation of Ir-Ir dimers following photoexcitation. Evolution of the Ir-Ir dimers (blue lines) and non-dimerized Ir-Ir pairs (red lines).