Probing the electronic topological transitions of WTe₂ under pressure using ultrafast spectroscopy

The differential reflectivity (DR) transients, $\Delta$R/R₀, ($\Delta$R=R-R₀) as a function of pump-probe delay under selected pressures are presented in Fig. 1(c), where the R or R₀ is the reflectivity with and without the pump, respectively. More pressure-dependent data are provided in Fig. S4 in the Supplemental Material. In the measurement [Fig. 1(c)], the polarization of the probe light was set as parallel in either an a direction (upper panel) or b direction (lower panel). We determined the crystalline orientation of the Td-WTe₂ crystal based on the angle-resolved Raman spectroscopy and fs-OPPS (see Figs. S1 and S3 in the Supplemental Material). Hereafter, these two probe-polarization geometries are referred to as parallel and perpendicular configurations. At ambient conditions, the DR signal features an initial instantaneous dropping, which is followed by a relaxation process composed with a fast rising and a subsequent slow decay superimposed with periodic oscillations. The periodic oscillation stems from the excitation of the CP corresponding to the A₁ optical phonon mode, with a frequency of 0.25 THz.[43, 44] The incoherent electronic response contains the fast and slow relaxation stages; the fast component is attributed to the energy dissipation of the photocarrier through optical phonon scattering, while the slow component should be governed by the interactions between the electrons and acoustic phonons[19] or the phonon-assisted e-h recombination between the electron and hole pockets,[45] exhibiting a long-lived offset within our time-scanning range.

As revealed by Figs. 1(c) and S4, the photocarrier dynamics in WTe₂ shows obvious in-plane anisotropy due to the directional one-dimensional tungsten-chain structure, which is akin to the observations in black phosphorous.[46] Clearly, the DR transients strongly depend on the external pressure. In addition, the two common pressure effects for both parallel and perpendicular polarization configurations can be directly identified from the Figs. 1(c) and S4. First, in both polarization configurations, the amplitude of the fast component with negative sign in the incoherent DR signal pronouncedly declines upon compressing, and almost disappears as the pressure is beyond 6 GPa. Instead, with further elevating pressure, the fast relaxation component over subpicosecond timescale resurge in the manner of rapid rising with positive sign above 15.5 GPa. Second, the CP oscillations suddenly vanish as the pressure is beyond 3.5 GPa. Previous works based on the high-pressure Raman and XRD experiments have corroborated that the adjacent Te-W-Te layers undergo an opposite sliding, leading to a structural transition from the orthorhombic Td-WTe₂ to monoclinic 1T’-WTe₂.[25, 26, 34] However, no consensus was reached as for the turning pressure of the Td-1T’ phase transition pressure. The reported transition points are scattered in the pressure ranges from 4.5 to 9.6 GPa.[25, 26, 34, 37, 39] The suppression of the CP excitation of the shear mode was proposed to serve as an indicator for Td-1T’ phase transition in MoTe₂,[47] although it was not accessed yet under high pressure. Therefore, we suggest that the disappearance of the CP oscillations at 3.5 GPa marks the onset of the Td-1T’ transition.

To gain the quantitative insights into the pressure dependence of the photocarrier dynamics, we fit the DR curve in terms of a damped harmonic oscillator function combined with a single-exponential function for each pressure, as follows:

where A_CP is the CP amplitude, $\tau$_CP is the lifetime of CP, f is the CP frequency, and $\varphi$ is the initial phase of CP oscillations. A_f and $\tau$_f denote the amplitude and time constant of the fast decay component, respectively. According to the two-temperature model,[43, 48] the e-ph coupling constant, $\lambda$, is roughly proportional to the energy relaxation rate 1/$\tau$_f. Thereby, first of all, we pay special attention to the $\tau$_f to assess the pressure evolution of the e-ph interactions. The extracted time constant for the fast decay process ($\tau$_f), for parallel and perpendicular configurations, versus pressure is presented in Fig. 1(d). As the probe light is along the a direction, the $\tau$_f monotonously decreases with increasing pressure, while shows an abrupt discontinuous change of approximately 37$\%$ in magnitude around 3.5 GPa. With further compression, the $\tau$_f again reduces with pressure up to 6 GPa, followed by a gradual rising at higher pressures. Interestingly, the pressure dependence of $\tau$_f obtained in the perpendicular configuration exhibits a different pressure behavior. The $\tau$_f essentially remains constant below 3.5 GPa, with an exception of a recognizable sudden drop at around 0.8 GPa. An abrupt rising of the $\tau$_f around 3.5 GPa is also discernible although the value is far smaller than that in the parallel configuration. Under higher pressure, the time constant pronouncedly increases up to 6 GPa, and subsequently dramatically shortens.

Overall, we clearly distinguished the three discontinuities around 0.8, 3.5, and 6 GPa for the pressure dependence of $\tau$_f, emphasizing the pressure-induced abruptions in the e-ph interactions. To further reveal if there are additional lattice structure changes except the Td-1T’ phase transition, we carried out the high-pressure Raman scattering measurement with short pressure step. As exemplified in Fig. S2 in the Supplemental Material, the pressure behaviors of Raman modes do not manifest any discontinuity but a subtle uptrend on the slop of the frequency at around 3.5 GPa in our studied pressure range, which is in line with earlier reports.[25, 26, 34] In addition, we can see that the CP experiences an initial increase in the amplitude before it completely disappears as shown in Fig. S4. The totally suppression of the CP signal occurs in a sharp pressure range as 0.5 GPa, which is in striking contrast with the reported gradual evolution from Td to 1T’ phase.[25, 26, 37] Thus, we infer that the abrupt changes in electronic structure, such as the Lifshitz transition should be taken into account to explain the anomalies of the pressure dependences of the photocarrier dynamics.

To unveil the pressure effects of the electronic structures for Td-WTe₂, we performed the DFT calculations, (see the details in the Supplemental Material). The calculated energy bands under pressure are shown in Fig. 2(a). Accordingly, the cross sections of the Fermi surface around the Fermi level at 0, 2.5, and 4 GPa were extracted, shown in Fig. 2(b). One can find that there exist sharp changes around Fermi level (-0.1 eV$\sim$0.1 eV) along $\Gamma$-Y with increasing pressure. At the pressure of 0 GPa, the type-II Weyl points, derived from the tiny band crossing above the Fermi level at around 0.05 eV, can be recognized, which is consistent with a previous report.[24] As the pressure increases up to about 2.5 GPa, both the electron and hole pockets gradually grow in size along with increasing pressure, and eventually the two electron pockets merge into one, as shown in Fig. 2(b). Such a topological change of the Fermi surface has been observed by the magneto-transport methods,[38] demonstrating a Lifshitz transition. The growth of the pocket volumes implies an augment of the density of electronic states at k-space around the energy-band extreme points, which can effectively render more available phase space at low-energy range for hot carrier distribution and thus shorten the fast decay time constant.[21, 22] Remarkably, the electronic density of states (DOS) along the $\Gamma$-X direction in k-space (along the b-axis in real space) surges due to the merging of the electron pockets, and thus can lead to a sudden drop of the $\tau$_f as the probe polarization is along the b-axis. Note that the $\tau$_f is not sensitive to the adjustment of the pump polarization, as shown in Fig S3(a) in the Supplemental Material. This observation seems plausible because the initial orientating momenta of the photocarriers, endowed by the polarized pump photon upon photoexcitation, would be rapidly randomized over a few tens of femtoseconds (shorter than the pulse duration of the fs laser used in our experiment) through carrier-carrier scatterings.[49]

Further increasing the pressure up to 4 GPa, our calculations reveal that a new Weyl point with linear band dispersion emerges at around the Fermi level, accompanying with a gap closing. At further compression, this new Weyl point vanishes, and a gap reopens but the Fermi surface topology retains when the pressure significantly increases up to about 6 GPa. This new Weyl point is quite different from that at low pressure (0$\sim$1 GPa). The initial eight Weyl points originate from the double degeneracies due to the spin-orbit coupling,[24] while the high-pressure Weyl points result from the point contact between the conduction and valence bands modulated by the external pressure. This result implies possibly distinct transport or ultrafast dynamical behaviors at different pressures. Since the e-ph interaction for the electronic band with a linear energy dispersion is commonly expected to be weakened due to the large Fermi velocity and small effective mass,[50] we propose that the formation of Weyl point can facilitate the decoupling between electron and phonon, resulting in the sudden increase of $\tau$_f around 3.5 GPa. The anisotropy of the new-born Weyl points in k-space should be responsible for the polarization sensitivity of the $\tau$_f. Especially, the $\tau$_f exhibits a reversed pressure behavior above 3.5 GPa for parallel and perpendicular configurations (Fig. 2(d)). Such an effect is consistent with the shift of the new Weyl points from the highly symmetrical $\Gamma$-Y direction (a-axis) to the $\Gamma$-X direction (b-axis) as unveiled by the calculation. We should mention that our calculations do not reveal sudden change in the electronic structure around 6 GPa for WTe₂ with Td phase, and cannot account for the turning point of 6 GPa of the pressure dependence of the $\tau$_f. We propose that there are two possible mechanisms underlying the anomaly. (1) The gradual Td-1T’ phase transition may lead to the hybrid phases with Td and 1T’ structural WTe₂ in the 4-6 GPa pressure range. This anomaly in photocarrier dynamics can be trigged when the 1T’ phase dominates with pressure. (2) Additional ETT likely takes place for 1T’ phase at around 6 GPa, resulting in the abruption of e-ph interaction. Overall, the anomalous transitions of $\tau$_f with pressure should be correlated with the electronic structure transition, the first anomaly around 0.8 GPa is associated with the Lifshitz transition of the electronic bands, and the sudden changes above 3.5 GPa has a close correspondence with the emergence of the new Weyl point at the Fermi level.

The pressure induced change in the electronic structure not only modifies the electronic response of the DR signal but also impacts the CP excitation. In Fig. 3(a), we plot the CP time-domain intensity profiles at different pressures as the color map, where the incoherent electronic background under different pressures was subtracted. Accordingly, the dynamical parameters including the frequency (f_CP), lifetime ($\tau$_CP), amplitude (A_CP), and phase ($\varphi$) versus the pressures are illustrated in the Figs. 3(b)-(e). Apparently, the CP frequency manifests blueshift with pressure, which is due to the pressure-induced enhancement of interlayer interactions. Moreover, we can see that the lifetime of CP shortens with increasing pressure, emphasizing the reinforcement of anharmonic interactions between phonons.[51] Such increase in an anharmonic coupling cannot fully account for the sudden vanishing of the CP oscillation above 3.5 GPa since the $\tau$_CP should not be zero albeit it reduces by a factor of 5. Indeed, the lifetime of CP is expected to be about 40 ps at 3.5 GPa according to the trend. In addition, the CP amplitude exhibits a prominent increase before it completely disappears as shown in Fig. 3(d). The sudden changes of the lifetime and amplitude of the CP with pressures further emphasize that the topological transition of the electronic structures plays nontrivial roles for the e-ph coupling.

As shown in Fig. 3(e), the phase $\varphi$ at ambient pressure is $\sim$0, indicating a cosine-type CP oscillation, which highlights that the coherent shear phonon stems from the displacive excitation.[52, 53] The displacive CP excitation is associated with the shift of the potential energy surface induced by the nonradiative energy delivery through the e-ph coupling.[54] Thereby, we suggest that the sudden changes of CP oscillation around 3.5 GPa reflect an abrupt reduction in the e-ph coupling. It can be seen that the phase shift rapidly increases with elevating pressure and amounts to -45^∘ around 3.5 GPa. In other word, the lattice response shows an extra delay in response to the photoexcitation under pressure. As for the displacive CP, if the lifetime of CP is far smaller than that of electronic relaxation, the phase is roughly linked with the frequency and CP lifetime, that is, $\varphi$$\approx$arctan(-1/(f$\tau$_CP )).[55] Accordingly, the maximum change in the phase is estimated about 3.2^∘ under pressure, which is much less than the experimental result. Previous work has revealed that the electron temperature and the carrier cooling during the nascent stage of electronic relaxation strongly impact the coordinate position of the potential energy surface of the excited state and can give rise to the large phase shift of CP in TiO₂.[56] Along this line, we suggest that an extraordinary large phase shift has a close correlation with the changes in the electronic structure. As shown by the above calculation (Fig. 2), both the DOS and the size of the Fermi pockets increase remarkably with pressure, allowing for more state occupations at the lower energy side. As a result, the initial carrier temperature upon photoexcitation is expected to be decreased along with the ETT under pressure. Such attenuation in electron temperature may lead to a non-instantaneous driving force for CP excitation.[56] Furthermore, we propose that the reduction both in the electron temperature and e-ph coupling due to the ETT and the formation of the Weyl point above 4 GPa decrease the energy exchange rate between electron and lattice, and further suppress the excitation of CP.

In summary, we employ the fs-OPPS in combination with DAC to investigate the pressure evolutions of nonequilibrium photocarrier dynamics in WTe₂, reflecting the details of the electronic structure over large k-space previously inaccessible under pressure. We find that the DR curves comprised with incoherent electronic response and CP manifest the anomalies around 0.8, 3.5, and 6 GPa, depending on the light polarization. With increasing pressure, the CP stemming from the A₁ mode starts to disappear above 3.5 GPa. Concomitantly, the fast time constant of the electronic relaxation shows a recognizable sudden drop around 0.8 GPa, and also a remarkable jump at around 3.5 GPa. Such pressure behaviors of photocarrier dynamics signify the electronic structure transitions in the crystal structure. The first-principles calculation reveals a topology change of Fermi surface and a new type-II Weyl energy dispersion below 6 GPa. Our work does not only develop the better understandings of the e-ph interactions in WTe₂, but also proposes a new technique to probe the electronic topological changes under pressure.