⇦Chinese Physics Letters, 2023, Vol. 40, No. 11, Article code 114202 Giant Nonlinear Optical Response in Topological Semimetal Molybdenum Phosphide Kai Hu (胡凯)1, Yujie Qin (秦羽婕)1, Liang Cheng (程亮)1, Youguo Shi (石友国)2, and Jingbo Qi (齐静波)1* Affiliations 1State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China 2Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Received 29 July 2023; accepted manuscript online 28 September 2023; published online 18 October 2023 *Corresponding author. Email: jbqi@uestc.edu.cn Citation Text: Hu K, Qin Y J, Cheng L et al. 2023 Chin. Phys. Lett. 40 114202    Abstract Nonlinear optical properties are investigated using the static and time-resolved second harmonic generation in the topological material molybdenum phosphide (MoP) with three-component fermions. Giant second harmonic generation signals are detected and the derived $\chi^{(2)}$ value is larger than that of the typical electro–optic material. Upon optical excitation, no photoinduced change of the symmetry is detected in MoP, which is quite different from previous observations in several other topological materials.
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 DOI:10.1088/0256-307X/40/11/114202 © 2023 Chinese Physics Society Article Text Over the recent 15 years, intensive investigations have been conducted on topological materials due to their exotic physical properties. This interest is particularly related to the special band topology of their electronic structures.[1-4] Theoretical predictions and experimental discoveries have revealed a series of such materials, including topological insulators, Dirac semimetals, and Weyl semimetals.[5-8] In these materials, novel optical properties under both equilibrium and nonequilibrium conditions have received special attention due to their potential applications in optoelectronics.[9-11] In topological systems, nonlinear optical responses are closely related to the correlated nontrivial electronic structures and crystal symmetry.[12] Thus, they naturally provide an effective way to probe the underlying physical properties and their relationships.[13,14] Specifically, the second harmonic generation (SHG) technique has been used to study the surface state in topological insulator Bi$_2$Se$_3$.[15] In the Weyl semimetal TaAs, SHG signals have revealed a colossal second-order nonlinear coefficient $\chi^{(2)}$ and optical-induced symmetry breaking.[14,16] Recently, researchers discovered new quasiparticle excitations in molybdenum phosphide (MoP) that go beyond the conventional Dirac–Weyl–Majorana classification.[17] These excitations are three-component fermions, protected by a combination of mirror and rotational symmetry, which differ from Weyl fermions. As a result, the band topology and crystal symmetry of MoP may lead to new optical phenomena. However, despite such potential, the nonlinear optical responses in MoP remain elusive. To address this gap in knowledge, we utilize both static and time-resolved SHG techniques to investigate the (quasi)equilibrium and dynamical properties of MoP. Our results show giant SHG signals corresponding to a significant $\chi^{(2)}$ value, larger than that of GaAs. Furthermore, we do not observe any transient changes in symmetry upon photoexcitation via monitoring the evolution of the anisotropic SHG patterns. This finding is a sharp contrast to previous reports on TaAs, ZrTe$_5$, and MoTe$_2$.[16,18,19]
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Fig. 1. (a) Schematic experimental setups used for the static and time-resolved SHG measurements. (b) Crystal structure of MoP. Blue and orange balls represent the molybdenum and phosphide atoms, respectively. $M_y$ and $M_z$ are the symmetry planes of MoP. The red dashed line represents the $C_{3z}$ rotational symmetry of MoP.
The experimental setups used in this study are shown in Fig. 1(a). Detailed information about the sample growth and experimental setups are given in the Supplementary Materials. In order to study the ultrafast dynamics of MoP, a femtosecond laser was employed as the incident pump or probe beam in the experiments. For the static SHG measurement, the femtosecond laser was normally incident to the (001) surface of MoP and the generated SHG signals were detected by a photomultiplier tube (PMT). To obtain the dependence of SHG on the polarization of incident light, a half-wave plate (generator) was placed before the sample and a polarizer (analyzer) was positioned in front of the PMT. The dependence of SHG on the sample's azimuthal angle could be obtained by simultaneously rotating the generator and the analyzer, which was equivalent to rotating the sample. We note that in MoP, the optical responses only come from a thin layer of $\sim$ 36 nm, based on the penetration depth derived from the refractive index (see Fig. S1 in the Supplementary Material). The relationship between the SHG signals and the crystal symmetry is manifested by the second-order susceptibility tensor $\chi_{ijk}^{(2)}$ in the following equation: \begin{align} P_i(2\omega)=\chi_{ijk}^{(2)}(2\omega)E_j(\omega)E_k(\omega). \tag {1} \end{align} The space group of MoP is $P\bar{6}m2$, which lacks inversion symmetry. Our experimental configurations enable us to concentrate on the $C_{3z}$ rotation symmetry and the $M_y$ and $M_z$ mirror symmetries, as shown in Fig. 1. We focused the fundamental light along the crystal's $c$-axis (or $z$ direction in the lab frame). According to the crystal symmetry of MoP, only one non-vanishing element is allowed in the second-order susceptibility $\chi_{ijk}^{(2)}$: $\chi_{xxx}^{(2)}=-\chi_{xyy}^{(2)}=-\chi_{yxy}^{(2)}=-\chi_{yyx}^{(2)}$. In the measurement, we performed angular scans with the analyzer being either parallel or perpendicular to the generator. Based on Eq. (1) and the non-vanishing $\chi_{ijk}^{(2)}$, the SHG intensity $I^{2\omega}$ as a function of the sample azimuth angle $\theta$ can be expressed as \begin{align} &I_{\rm para}^{2\omega}(\theta) \propto |\chi_{xxx}^{(2)} \sin(3\theta)|^2, \notag \\ &I_{\rm perp}^{2\omega}(\theta) \propto |\chi_{xxx}^{(2)} \cos(3\theta)|^2. \tag {2} \end{align} The angular-dependent data are presented in Fig. 2(a). The results clearly show a six-fold angular dependence, with a 60$^\circ$ angle shift between the parallel and perpendicular configurations. These findings are in excellent consistency with our fitting using Eq. (2), as indicated by the solid lines. In order to extract the exact value of $\chi_{ijk}^{(2)}$ of MoP, we also repeated the experiments on ZnTe and TaAs, which have known $\chi_{ijk}^{(2)}$ values. Their angular-dependent data are shown in Figs. 2(b) and 2(c). The relationship between the SHG signals and $\chi_{ijk}^{(2)}$ is also analyzed based on their crystal symmetry. From calculations, we can obtain the SHG signals $I^{2\omega}$ as a function of $\chi^{(2)}$ for MoP, ZnTe and TaAs, \begin{align} I_{\rm para,MoP}^{2\omega}(\theta) \propto\,& \Big|\chi_{xxx}^{(2)}\sin(3\theta)\Big|^2 \notag \\ I_{\rm para,ZnTe}^{2\omega}(\theta) \propto\,& \Big|3\chi_{xyz}^{(2)}\sin(\theta)\cos(\theta)^2\Big|^2 \notag \\ I_{\rm para,TaAs}^{2\omega}(\theta) \propto\,&\Big|\cos(\theta)[-5+\cos(2\theta)][2\chi_{xxz}^{(2)}+\chi_{zxx}^{(2)}]\notag\\ &-2\cos(\theta)^3\chi_{zzz}^{(2)}\Big|^2. \tag {3} \end{align} Since we used the reflection geometry in the measurement, $\chi^{(2)}$ needs to be modified by a coefficient according to the method reported previously.[14,20] The real nonlinear susceptibility $\chi_{\scriptscriptstyle{\rm R}}^{(2)}$ can be expressed as \begin{align} &\chi_{\scriptscriptstyle{\rm R}}^{(2)}=\frac{\chi^{(2)}}{[\epsilon^{1/2}(2\omega)+\epsilon^{1/2}(\omega)] [\epsilon^{1/2}(2\omega)+1]}T(\omega)^2, \tag {4} \end{align} where $\epsilon$ represents the relative dielectric constant of each material, $T(\omega) = 2\cdot[n(\omega)+1]^{-1}$ is the Fresnel coefficient. The refractive index of MoP was measured using the Fresnel equation and is listed in the Supplementary Material. The refractive index of TaAs was extracted from Ref. [13]. Based on Eq. (3), the peak values of the angular-dependent SHG signals are located at different angles. Then, the second-order susceptibility of MoP is obtained by comparing its SHG intensity with that of its reference sample counterpart, ZnTe. In particular, the peak intensities in Fig. 2 are proportional to $|4\chi_{xxz}^{(2)}+2\chi_{zxx}^{(2)}+\chi_{zzz}^{(2)}|^{2}/27$ in TaAs, 4$|\chi_{xyz}^{(2)}|^2/3$ in ZnTe and $|\chi_{xxx}^{(2)}|^{2}$ in MoP, respectively. The ratios between these peak intensities in Fig. 2 are 2.14($\pm 0.2$) (MoP:ZnTe) and 4($\pm 0.31$) (TaAs:ZnTe). Further, we used 500 pm/V for the $\chi_{xyz}^{(2)}$ of ZnTe as Ref. [21]. Based on the above values, we can obtain $d_{33}$ ($=\chi_{zzz}^{(2)}/2$) of TaAs to be 3390($\pm 263$) pm/V, and $\chi_{xxx}^{(2)}$ of MoP to be 832($\pm 40$) pm/V. The former value is obtained using the complex refractive index measured in Ref. [13], and is smaller than that reported in Ref. [14], where the dielectric constant of TaAs was not provided. The latter value is also significant since it is larger than the $\chi^{(2)}$ of the nonlinear benchmark material GaAs ($\chi^{(2)}_{xyz}=700$ pm/V).[22]
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Fig. 2. Experimental SHG data of MoP (001), ZnTe (110), TaAs (112) with the same spot size. (a) SHG polar plot for MoP, showing six-fold angular dependence in both parallel and perpendicular configurations. The red and blue solid lines are the fitted results for parallel and perpendicular configurations, respectively. (b) SHG polar plot for ZnTe. The signals measured from parallel and perpendicular configurations exhibit different angular dependence, which are well fitted via Eq. (1). (c) SHG polar plot for TaAs. The signal measured in a perpendicular configuration is much smaller than that in the parallel one. This can be understood from the difference in various $\chi^{(2)}$ of TaAs. (d) Comparison between SHG signals of MoP and ZnTe measured in parallel configuration. The SHG intensity of MoP is much larger than that of ZnTe.
The large nonlinear optical response is very similar to the findings reported for Weyl semimetal TaAs.[14,23] Therefore, we believe that they share the same origin. According to the theory,[14,23] the large second-order susceptibility can be closely connected to the band structure topology and, remarkably, the presence of Weyl nodes. Such an exotic electronic structure also exists in MoP and includes not only the Weyl nodes but also the triply degenerate points. In experiments, the measured SHG signals have contributions from all optical transitions between the relevant energy levels, which apparently contain the transitions involving bands near these special points. Therefore, we attribute the giant $\chi^{(2)}$ value of MoP to its unique electronic structures. Despite the giant nonlinear response, we note that pairs of Weyl points also coexist with the three-component fermions in MoP.[17] In addition, several works on topological semimetals demonstrate that the original symmetry can be broken due to photoexcitation.[16,18,19] These two aspects inspired us to investigate the photoinduced dynamic properties of MoP. Therefore, we carried out time-resolved SHG experiments based on the optical pump-probe scheme, as shown in Fig. 1(a). In addition to the transient SHG signal change $\Delta I^{2\omega}$/$I^{2\omega}$ at 2$\hbar\omega$, for comparison purposes, we also measured the transient reflectivity change $\Delta R/R$ at the fundamental frequency $\hbar\omega$. Figure 3 shows the typical results of $\Delta I^{2\omega}$/$I^{2\omega}$ and $\Delta R/R$ measured at room temperature. Clearly, these two profiles in Fig. 3 are quite similar, i.e. the signals experience an initial instantaneous rise upon photoexcitation, followed by some decay processes thereafter. In order to investigate the relaxation dynamics quantitatively, we fit the signals before $\sim$ 20 ps using the following equation: \begin{equation} \frac{\Delta \gamma(t)}{\gamma}=\Big[\sum\limits_{j=3}A_je^{-\frac{t}{\tau_j}}\Big]\otimes G(t), \tag {5} \end{equation} where $\gamma = I^{2\omega}$ or $R$, and we have included three relaxation processes with amplitudes $A_j$ and decay times $\tau_j$ ($j=1,\,2,\,3$), respectively. Here, $A_1$ and $A_3$ have opposite signs to that of $A_2$. $G(t)$ is the Gaussian function representing the correlation between the pump and probe pulse. Figure 3 shows our fitted data, which agree well with the experimental results. Specifically, three distinct relaxation processes can be clearly extracted. Interestingly, both of the ultrafast dynamics measured by the reflectivity and SHG exhibit the same characteristics, which implies that their relaxation processes may have similar physical origins. We note that the bump-like feature around 5 ps in the $\Delta R/R$ signal was attributed to some oscillatory behavior by a previous work.[24] It seems that our results do not support this claim (see additional data in Fig. 5 and in the Supplementary Material).
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Fig. 3. Transient optical response in MoP. (a) The variations of SHG signals as a function of time measured at pump fluence of 1.6 mJ/cm$^{2}$. Using Eq. (5), we can fit the curve with three different relaxation processes: $\tau_1$, $\tau_2$, $\tau_3$. The fitted results are shown by the red solid lines. (b) The transient reflectivity change measured under the same conditions. Similarly, the curve can also be fitted with the three decay processes as shown by the solid lines.
In order to study the relaxation processes, we further measured $\Delta R/R$ as a function of temperature.[25,26] In the experiment, a small pump fluence of 0.5 µJ/cm$^2$ was used. Figure 4(a) shows the typical measured $\Delta R/R$ data within 20 ps. Signals in a long timescale can be found in the Supplementary Material. The extracted $\tau_1$ and $\tau_2$ illustrate an increasing tendency as the temperature increases, as shown in Figs. 4(b) and 4(c). In order to analyze the data, we use a model that explains the ultrafast decay process by the energy exchange between the electron and lattice subsystems.[27,28] As shown in Fig. 4, the fitting results agree quite well with the experimental data. Specifically, $\tau_1$ has a sub-ps timescale, which is usually due to the scattering between photo-excited carriers and optical phonons. Since $\tau_2$ is of ps timescale, it may indicate that this relaxation channel involves low-energy optical or acoustic phonons. Noticeably, $\tau_3$ has a timescale of tens to hundreds of ps, which conventionally arises from the thermal diffusion process. We can further achieve the electron–phonon coupling constants $g_{\infty}$ of $1.119\times10^{16}$ W$\cdot$m$^{-3}\cdot$K$^{-1}$ for the $\tau_1$ process. Furthermore, the electron–phonon coupling coefficient $\lambda\langle \omega^2\rangle$ can be evaluated by employing $\tau^{-1}=3\hbar\lambda\langle \omega^2\rangle(\pi k_{\scriptscriptstyle{\rm B}}T_{\rm e})^{-1}$,[28] which is $6.85\times 10^{24}$ Hz$^2$.
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Fig. 4. (a) Typical $\Delta R/R$ data measured at several temperature. [(b), (c)] The extracted $\tau_1$ and $\tau_2$ (scattered points) as a function of temperature, where the solid line represents the fitting results using the method in Ref. [27].
We further investigate if there is any broken symmetry within the system upon photoexcitation via measuring the angular dependence of the SHG signals at different time delays. As shown in Fig. 5(a), we selected four time points at different stages of the relaxation processes in the time domain. The corresponding $I^{2\omega}(t,\theta$) data are shown in Fig. 5(b). We can clearly observe that the SHG signals always exhibit six-fold symmetry at all the selected time delays. The result indicates that the symmetry property of MoP remains unchanged upon photoexcitation. This finding is different from the previous reports of Weyl semimetals such as TaAs and MoTe$_2$.[16,19] No photoinduced symmetry broken was observed in MoP even if the pump fluence had the same order as those used in other experiments.[19] This result can be attributed to two reasons. In TaAs, the pump-induced symmetry change is due to the asymmetry of carrier distribution induced by the colossal photocurrent involving the Weyl nodes near the Fermi surface. However, the triply degenerate points as well as the Weyl points in MoP are located at energies far away from the Fermi level,[17] which do not support the generation of a large photocurrent. In fact, according to Ref. [29], there is no detectable photocurrent in MoP under 800 nm excitation. Our observation, therefore, is consistent with this consideration and fact. In MoTe$_2$ with a layered structure, the electric field of laser pulses was found to be capable of accelerating the electrons in the valence band and could change the coupling strength between layers. As a result, a new equilibrium phase could be established by interlayer shear movement. However, this nonthermal mechanism will not happen in MoP since it has no layered structure. This is also consistent with our observations.
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Fig. 5. (a) The time delays chosen to measure the angular dependence of SHG in (b). (b) The angular dependence of SHG corresponding to the delays in (a). All of the plots exhibit similar patterns of six-fold angular dependence. Only the intensities of signals vary at different points; peak point 2 corresponds to the minimum value.
In summary, we have investigated the static and dynamic nonlinear optical responses in topological semimetal MoP. After careful investigations, colossal SHG signals are found in MoP and the corresponding $\chi^{(2)}$ is 832 pm/V, which is larger than that of GaAs. This giant nonlinear response may be related to the topological band structure of MoP, similar to TaAs. In addition, we also attempted to use an ultrafast laser to manipulate the symmetry of MoP. However, we did not observe any symmetry change based on the time-resolved SHG patterns. Overall, our measurements demonstrate that MoP has potential applications in nonlinear optics. Acknowledgments. We thank Dr. Hongming Weng for valuable discussion. This work was supported by the National Key R&D Program of China (Grant No. 2022YFA1403000), and the National Natural Science Foundation of China (Grant Nos. 11974070, 92365102, 11974306, 12034017, 11734006, and 11774401). References Observation of a large-gap topological-insulator class with a single Dirac cone on the surfaceWeyl and Dirac semimetals in three-dimensional solidsExperimental Discovery of Weyl Semimetal TaAsType-II Weyl semimetalsDiscovery of a Weyl fermion semimetal and topological Fermi arcsTopological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surfaceExperimental Realization of a Three-Dimensional Dirac SemimetalDiscovery of a Three-Dimensional Topological Dirac Semimetal, Na3BiPhotonic Floquet topological insulatorsUltrafast carrier and phonon dynamics in Bi2Se3 crystalsUltrafast carrier and lattice dynamics in the Dirac semimetal NiTe 2 Topological nature of nonlinear optical effects in solidsChiral terahertz wave emission from the Weyl semimetal TaAsGiant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetalsNonlinear Optical Probe of Tunable Surface Electrons on a Topological InsulatorPhotocurrent-driven transient symmetry breaking in the Weyl semimetal TaAsObservation of three-component fermions in the topological semimetal molybdenum phosphideA light-induced phononic symmetry switch and giant dissipationless topological photocurrent in ZrTe5Light-Induced Subpicosecond Lattice Symmetry Switch in MoTe 2 Light Waves at the Boundary of Nonlinear MediaDispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnSSecond-Harmonic Generation in GaAs: Experiment versus Theoretical Predictions of χ x y z ( 2 ) Resonance-enhanced optical nonlinearity in the Weyl semimetal TaAsTime-resolved ultrafast dynamics in triple degenerate topological semimetal molybdenum phosphideHybridization Dynamics in CeCoIn 5 Revealed by Ultrafast Optical SpectroscopyExtremely low-energy collective modes in a quasi-one-dimensional topological systemFemtosecond spectroscopy of electron-electron and electron-phonon energy relaxation in Ag and AuTheory of thermal relaxation of electrons in metalsGeneration and control of photo-excited thermal currents in triple degenerate topological semimetal MoP with circularly polarized ultrafast light pulses
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