Abstract
Harmonic generation is a general characteristic of driven nonlinear systems, and serves as an efficient tool for investigating the fundamental principles that govern the ultrafast nonlinear dynamics. Here, we report on terahertzfield driven highharmonic generation in the threedimensional Dirac semimetal Cd_{3}As_{2} at room temperature. Excited by linearlypolarized multicycle terahertz pulses, the third, fifth, and seventhorder harmonic generation is very efficient and detected via timeresolved spectroscopic techniques. The observed harmonic radiation is further studied as a function of pumppulse fluence. Their fluence dependence is found to deviate evidently from the expected powerlaw dependence in the perturbative regime. The observed highly nonperturbative behavior is reproduced based on our analysis of the intraband kinetics of the terahertzfield driven nonequilibrium state using the Boltzmann transport theory. Our results indicate that the driven nonlinear kinetics of the Dirac electrons plays the central role for the observed highly nonlinear response.
Introduction
In atomic gases^{1}, highharmonic radiation is produced via a threestep process of ionization, acceleration, and recollision by a strongfield infrared laser. This mechanism has been intensively investigated in the extreme ultraviolet and soft Xray regions^{2,3}, forming the basis of attosecond research^{1}. In solidstate materials, which are characterized by crystalline symmetry and strong interactions, yielding of harmonics has just recently been reported^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}. The observed highharmonic generation was interpreted with fundamentally different mechanisms, such as interband tunneling combined with dynamical Bloch oscillations^{4,5,7,8,9,10,11,12,21,22}, intraband thermodynamics^{16} and nonlinear dynamics^{23}, and manybody electronic interactions^{6,15,17,18,19,24}. Here, in a distinctly different context of a threedimensional Dirac semimetal, we report on experimental observation of highharmonic generation up to the seventh order driven by strongfield terahertz pulses. The observed nonperturbative highharmonic generation is interpreted as a generic feature of terahertzfielddriven nonlinear intraband kinetics of Dirac fermions. We anticipate that our results will trigger great interest in detection, manipulation, and coherent control of the nonlinear response in the vast family of threedimensional Dirac and Weyl materials.
Highharmonic generation (HHG) in twodimensional Dirac semimetals (singlelayer graphene^{14,16,17} and 45layer graphene^{7}) has been reported very recently for pump pulses both in the terahertz (10^{12} Hz, 1 THz ∼4 eV)^{7,16} and midinfrared or nearinfrared (0.2–0.8 eV) ranges^{14,17}. Although previous theoretical investigations pointed out that the peculiar linear energymomentum dispersion relation (Dirac cone) should be essential for HHG in graphene (see e.g., ref. ^{25,26,27}), the strong dependence on pump laser frequencies observed in the experiments favors different mechanisms. For the midinfrared or nearinfrared HHG, the interband transitions (combined with Bloch oscillations) play the crucial role, while the linear dispersion relation is not a prerequisite^{14}. A similar mechanism involving interband transitions can also be applied to terahertz (THz) HHG in lightlydoped multilayer graphene, whereas the exact shape of the carrier distribution was found to play only a minor role^{7}. In contrast, for heavily electrondoped graphene, intraband processes become important and HHG was ascribed to THzfield heated hotelectrons while assuming the electron subsystem thermalized quasiinstantaneously^{16}.
One may expect to observe THz HHG universally in the Dirac materials also of higher dimension, e.g., threedimensional (3D) Dirac or Weyl semimetals. However, THz HHG so far has not been reported for this class of materials, and the mechanism for observing THz HHG in a 3D Dirac material remains elusive. Here, we report on timeresolved detection of nonperturbative THz HHG in the 3D Dirac semimetal Cd_{3}As_{2}, and a realtime theoretical analysis of the THzfield driven kinetics of the Dirac fermions that is directly linked to the linear dispersion relation. Our results show that the THzfield driven nonlinear kinetics of the Dirac electrons is the mechanism responsible for the efficient generation of highharmonic radiation, as well as for its nonperturbative fluence dependence in Cd_{3}As_{2}.
Results
Third harmonic generation
As being both theoretically predicted and experimentally confirmed^{28,29,30,31,32,33}, Cd_{3}As_{2} is a wellestablished roomtemperature 3D Dirac semimetal with Fermi velocity about 10^{5} to 10^{6} m/s. Very compelling topological properties such as topological surface states and 3D quantum Hall effects have been realized in this system^{34,35,36,37,38}. In highquality Cd_{3}As_{2} thin films prepared by molecular beam epitaxy^{39}, we observe HHG unprecedentedly up to the seventh order in the nonperturbative regime. THz harmonic radiation was recorded with femtosecond resolution at room temperature. Figure 1a displays the detected electric field as a function of time delay for the third harmonic radiation, induced by a multicycle pump pulse (Fig. 2a) with a peak field of 144 kV/cm characterized by its central frequency of f = 0.67 THz (Fig. 1b). The power spectrum of the harmonic radiation is obtained by Fourier transformation of the timedomain signals, which exhibits a sharp peak at 3f = 2.01 THz (Fig. 1b). The intensity of the harmonic radiation is nearly independent on the polarization of the pump pulse within the sample surface (see Supplementary Fig. 2). To further characterize the third harmonic generation, we measured the timeresolved signals for different pumppulse intensities. As summarized in Fig. 1c, the fluence dependence of the third harmonic radiations remarkably does not follow the cubic law, but exhibits a powerlaw dependence as I_{3f} ∝ I^{2.5}_{f} on the pumppulse intensity I_{f}, which reveals a nonperturbative nonlinear response.
THz driven nonlinear kinetics
To understand the nonperturbative harmonic generation, we performed realtime theoretical analysis of the THz driven kinetics of the 3D Dirac electrons. For the electrondoped system, interband electronic excitations are Pauliblocked for onephoton transitions in the THz frequency range, thus we focus on the intraband kinetics of the nonequilibrium state by adopting a statistical approach of the Boltzmann transport theory. The initial state of thermodynamic equilibrium is defined by the roomtemperature FermiDirac distribution \(f_0 \left[ \it{\epsilon}\left( {\mathbf{p}} \right) \right] = \left[1+e^{\frac{\it{\epsilon }\left( {\mathbf{p}} \right)\it{\epsilon }_F}{k_BT}} \right ]^{1}\) for the 3D Dirac electrons obeying the linear dispersion relation \({\it{\epsilon }}\left( {\mathbf{p}} \right) = v_{\mathrm{F}}\left {\mathbf{p}} \right\), with p and v_{F} denoting momentum and Fermi velocity, respectively, ε_{F} for Fermi energy, k_{B} the Boltzmann constant, and T for temperature. In presence of the THz pulse, the driven transient state is characterized by the distribution function f(t, p), the timedependent evolution of which is governed by the Boltzmann equation^{40,41}
where the linear dispersion relation has been implemented, e and E(t) denote the electron charge and the THz electric field, respectively, and τ is the characteristic relaxation time for intraband processes, which is a phenomenological parameter (see “Methods”). In particular, we do not presume that the electron subsystem thermalizes quasiinstantaneously or a FermiDirac distribution should be obeyed by the transient states. In contrast, by solving the Boltzmann equation, we obtain the realtime distribution of the transient state. By comparing it with the equilibriumstate FermiDirac distribution, we can claim whether the corresponding transient state is nearly thermalized or far from thermodynamic equilibrium. Furthermore, we can derive the timedependent current density, hence the THz fieldinduced harmonic radiation, the fluence dependence of which can be compared to the experimental observations.
For the experimentally implemented THz pump pulses (see Fig. 2a for the waveform) with a typical electricfield peak strength of 110 kV/cm, the obtained current density (Fig. 2b) and transientstate distribution functions are illustrated in Fig. 2c–f, corresponding to the representative time delays (red symbols) marked in Fig. 2a, b, for the experimental values of Fermi energy ε_{F} = 118 meV and Fermi velocity v_{F} = 7.8 × 10^{5} m/s as estimated from Shubnikovde Haas oscillations^{39}, and the relaxation time τ = 10 fs. The electric field of the linearlypolarized pump pulse is set along the p_{z} direction.
The microscopic origin of HHG resides in the nonlinear kinetics of the electron distribution (see Fig. 2d–f and Supplementary Fig. 5) combined with the linear energy–momentum dispersion relation. Before the pump pulse arrives, the electrons in the upper band are in thermodynamic equilibrium, and fill the Diraccone up to around the Fermi energy according to the FermiDirac distribution (Fig. 2c). When the pump pulse is present, the electrons are not only accelerated by the THz electric fields, but at the same time also scattered. Although the latter process is dissipative, the former one can very efficiently accumulate energy into the electron subsystem, leading to a stretched and shifted distribution along the field. In particular, at the peak field (symbol point 2 marked in Fig. 2a), the distribution is most strongly stretched and shifted in the field direction (Fig. 2d) resulting in the maximum current density and a peculiar flatpeaklike feature (Fig. 2b), thereby leading to very efficient HHG. In clear contrast to the FermiDirac distribution of a thermodynamic equilibrium state that is spherically symmetric for the 3D Dirac electrons (manifested as circularly symmetric in the 2D plots); the obtained strongly stretched and highly asymmetric distribution due to the presence of the strong THz field evidently shows that the electron subsystem is far from thermodynamic equilibrium. As shown in Fig. 2f, the electron distribution becomes nearly symmetric in low THz fields, indicating that a quasithermalized situation is reconciled in the lowfield limit.
For various pumppulse peak field strength, the intensity of the thirdharmonic radiation is shown in Fig. 1c for relaxation time τ = 10 fs. The peak field strength in the sample is estimated as the average value over the film thickness. The theoretical results reproduce excellently the observed nonperturbative fluence dependence of the thirdharmonic generation up to about 80 kV/cm of the peak field strength, though a deviation from the experimental data occurs at higher fluences. This deviation could be due to enhanced probability of interband multiphoton tunneling in the high electricfield limit, which is not included in our semiclassical analysis. Nevertheless, we found that the nonperturbative dependence on pumppulse fluence is a generic feature of the THz driven nonequilibrium states in the Dirac semimetals. Furthermore, we found that efficiency and fluencedependence of the THz HHG is very sensitive to the scattering rate 1/τ. By decreasing the scattering rate (or suppressing the dissipative processes), the transient distribution function is further stretched for the same electricfield strength, resulting in greater current density (c.f. τ = 30 fs in Fig. 2b) and enhanced THz HHG (see Supplementary Fig. 3, Supplementary Fig. 4, Supplementary Fig. 5, and Supplementary Movie 1 Supplementary Movie 2 for the realtime evolution of the distribution driven by the THz pulse in Fig. 2a). Our theoretical calculations further reveal that for a fixed scattering rate the harmonic generation is enhanced at a higher Fermi energy (see Supplementary Fig. 6), which is compelling for further experimental studies.
Higherorder harmonic generation
In order to detect higherorder harmonic radiation, we utilized lowerfrequency and strongfield THz pump pulses (see “Methods”)^{42}. Figure 3a shows the observed harmonic radiation up to the seventh order for the pumppulse frequency of 0.3 THz (see Fig. 3b for the waveform). Only the oddorder harmonics are observed, providing the spectroscopic evidence for the existence of inversion symmetry in the crystalline structure of Cd_{3}As_{2} (see ref. ^{33}). Our experimental results not only set the record for THz HHG in the 3D Dirac materials, but also present the striking observation of the nonperturbative fluence dependence for all the observed harmonic orders, as presented in Fig. 3c–e.
For the third harmonic radiation, the fluence dependence is also slightly below the cubic powerlaw dependence, similar to the behavior for the 0.7 THz pump pulse. Moreover, for the higherorder harmonics, the deviation from the corresponding perturbative powerlaw dependence is further increased. These features are perfectly captured by our quantitative theoretical analysis. By implementing the experimental pump pulse (see Fig. 3b) in our calculations, the timeresolved harmonic signals are derived as a function of pumppulse fluence. The best fitting for all the experimentally observed HHG is achieved at τ = 10 fs (see Fig. 3c–e). The obtained value of τ = 10 fs is comparable to that in graphene as directly obtained via timeresolved and angleresolved photoemission spectroscopic measurements^{43}. While such measurements have not been reported in Cd_{3}As_{2}, an estimate based on the Shubnikovde Haas measurements provides a τ value of the same order^{39}. These results strongly indicate that the THz fielddriven nonlinear kinetics of the Dirac electrons is the mechanism responsible for the observed nonperturbative nonlinear response in Cd_{3}As_{2}. Although for the seventh harmonic the experimental uncertainty is enhanced at the lowest fluence, the fluence dependence far away from the perturbative one is a clear and consistent experimental and theoretical observation. The nonperturbative response could be qualitatively understood in a way that the effective nonlinear susceptibilities are also function of the THz field due to the higherorder nonlinear response. We note that the observed nonperturbative response suggests that the experimental setting is close to but still below the socalled highharmonic plateau regime, in which the HHG intensity remains almost constant for the high orders and drops abruptly at a cutoff frequency as found in gases as well as in solids^{1,20}.
Discussion
The established mechanism of THz HHG here based on the driven nonlinear kinetics of Dirac electrons is different from those mechanisms proposed for HHG in graphene^{7,14,16,17}, in which either the interband transitions were found playing the dominant role or the intraband electron subsystem is assumed to thermalize quasiinstantaneously. In contrast, in the context of the 3D Dirac system, we found that, firstly, in the presence of strong THz fields, the entire intraband distribution is strongly stretched and highly asymmetric, denying a description using the FermiDirac distribution of thermodynamic equilibrium states that is symmetric along the Dirac cone. Secondly, for the intraband kinetics, the linear energymomentum dispersion is crucial for the THz HHG, whereas for a parabolic dispersion in the singleparticle picture, the induced radiation field \(E_{{\mathrm{out}}} \propto \frac{{{\mathrm{d}}j}}{{{\mathrm{d}}t}} \propto \frac{{{\mathrm{d}}v}}{{{\mathrm{d}}t}} \propto E_{{\mathrm{in}}},\) should follow the pump field E_{in}, hardly yielding harmonics. Thirdly, the exact shape of the electron distribution and its realtime evolution, as obtained from the Boltzmann transport theory, is directly responsible for the THz HHG. A higher efficiency is revealed for the cases of a more strongly stretched and highly asymmetric distribution, due to stronger THz electric field and/or reduced scattering rate.
In conclusion, we have observed THz driven HHG up to the seventh order unprecedentedly in the 3D Dirac semimetal Cd_{3}As_{2}. The fluence dependence of all the observed HHG was found well beyond the perturbative regime. By performing realtime quantitative analysis of the THz fielddriven intraband kinetics of the Dirac electrons using the Boltzmann transport theory, we have established the nonlinear intraband kinetics as the mechanism for the observed THz HHG in Cd_{3}As_{2}. The mechanism found here for THz HHG is expected to be universal in the vast family of 3D Dirac and Weyl materials^{44}, which provides strategies for pursuing high efficiency of THz HHG, and establishes HHG as a sensitive tool for exploring the interplay of various degrees of freedom. Towards the high electricfield regime, an experimental realization of THz HHG plateau in the Dirac materials and a full quantummechanical dynamic analysis are still outstanding from both the fundamental and the application points of view. Recently, nonperturbative THz thirdharmonic generation in Cd_{3}As_{2} was also reported in Ref. ^{45}.
Methods
Terahertz spectroscopy
We performed terahertz THz HHG experiments with THz sources based on a femtosecond laser system and on a linear electron accelerator. For the former, broadband THz radiation was generated through tilted pulse front scheme utilizing lithium niobate crystal^{46,47,48}. With initial laser pulse energy around 1.5 mJ at 800 nm central wavelength and 100 fs pulse duration broadband THz radiation with up to 3 µJ pulse energy was generated. At the linear accelerator in Helmholtz Zentrum DresdenRossendorf, multicycle superradiant THz pulses were generated in an undulator from ultrashort relativistic electron bunches^{42}. The generated THz radiation is carrier envelope phase stable, linear polarized with tunable emitted radiation frequency. The accelerator was operated at 100 kHz and was synchronized with an external femtosecond laser system. The latter served as probe in electrooptical sampling. To achieve high level of synchronization, a pulseresolved detection scheme was employed^{49}. To produce narrowband THz radiation, corresponding bandpass filters were used (see Supplementary Fig. 1 for more information).
Sample preparation and characterization
Highquality thin films of Cd_{3}As_{2} were grown by PerkinElmer (Waltham, MA) 425B molecular beam epitaxy system^{39}. The substrate of freshcleaved 2inch mica (~70 µm in thickness) was annealed at 300 °C for 30 min to remove absorbed molecules. Then 10 nmthick CdTe was deposited as buffer layer before the Cd_{3}As_{2} growth. Cd_{3}As_{2} bulk material (99.9999%, American Elements Inc., Los Angeles, CA) was evaporated on to CdTe at 170 °C. The growth was in situ monitored by reflection highenergy electron diffraction (RHEED) system. The sample surface is parallel to the crystallographic (112) plane. Part of the sample was patterned in Hall bar geometry and performed magnetic resistance measurement on physical properties measurement system (PPMS) (Quantum Design Inc.). Fermi energy and Fermi velocity of the 120 nmthick Cd_{3}As_{2} samples was estimated as E_{F} = 118 meV and v_{F} = 7.8 × 10^{5} m/s from the Shubnikovde Haas oscillations. THz transmission was characterized in the linear response regime by standard electrooptical sampling scheme.
Kinetic theory
Our theoretical analysis employed a statistical approach of the semiclassical Boltzmann transport theory with an effective relaxation time^{40,41,50,51,51,53}. The semiclassical description of particles is captured by a single particle distribution function f(t, r, p) in phase space. Observables can be calculated as integrals over momentum space. In order to calculate f(t, r, p) one needs to solve the Boltzmann equation
The left hand side of this equation corresponds to the collisionless evolution in phase space. The collision integral can either be calculated perturbatively from scattering amplitudes or chosen phenomenologically. In this work we use phenomenological relaxation time approximation and choose Bhatnagar–Gross–Krook collision operator^{40}.
The explicit form of the Boltzmann equation follows from the (inverted) equations of motion for the electron’s wavepacket^{51,52,53}
with the electromagnetic fields E and B, the Berry curvature Ω, the Planck constant \(\hbar\), and the elementary charg e. ε_{k} denotes the dispersion relation and \(D = 1 + \frac{{\boldsymbol{e}}}{\hbar }{\mathbf{B}} \cdot {\mathbf{\Omega }}\) is the modified phase space volume element. For the linearly polarized THz pulses, we consider the dominant effects of the electric field while neglecting the magnetic field in our further analysis. Consequently, the (inverted) equations of motion take the following simple form
Since we are interested in a homogenous solution, only the equation for \({\dot{\mathbf{p}}}\) is incorporated in the Boltzmann equation. The equation for \({\dot{\mathbf{r}}}\) is used to define the current density as follows:
Nevertheless, it can be shown that the second term in this equation (proportional to E × Ω) does not contribute to j(t) in the case of linearly polarized THz pulse, corresponding to the present experimental setting. Therefore, for the particular experiment being reported now, we can write
For the THz frequencies in our experiments, interband electronic transitions are Pauliblocked for the electrondoped Cd_{3}As_{2} samples. Thus, to study the intraband electron dynamics, it is justified to adopt one relaxation scale. In addition to that the underlying impurities in the system can lead to nonconservation of charge and momentum. As a result, we expect that the collision integral of the following form
will correctly reproduce the experimental data. In equilibrium the distribution function should depend on collisional invariants
where \(\beta \equiv 1/k_{\mathrm{B}}T\) with the Boltzmann constant k_{B}, ε_{F} denotes the Fermi energy, and the linear dispersion relation \({\it{\epsilon }}_{\boldsymbol{p}} = v_{\mathrm{F}}\left {\mathbf{p}} \right\) of the Dirac material has been implemented. Finally, considering only homogeneous response, we arrived at the following Boltzmann equation
where the external driving force F = −eE is implemented for electrons moving in the THz electric field E. In order to solve this equation, we Fourier transform the distribution function \(f\left( {t,{\mathbf{p}}} \right) = \frac{1}{{2\pi }}{\int} {dz\tilde f( {t,p_{\mathrm{x}},p_{\mathrm{y}},z})} {\mathrm{exp}}\left( {izp_{\mathrm{z}}} \right)\), which gives an ordinary differential equation
where the electric field E has been set along the z direction.
The ordinary differential equation is solved numerically with the experimental THz fields as an input. Having the distribution function, we calculate its moments to get current density. The expression for current density has the following form
where \({\hat{\mathbf{p}}}\) denotes the unit vector along the momentum direction. The relation between the induced current and the external oscillating field serves as the basis for analysis of higherharmonic generation.
Data availability
Data supporting the findings of this work are available from the corresponding authors upon reasonable request. Further requests on the raw presorted and prebinned data should be sent to HZDR via S.K.
Code availability
Computer code supporting the findings of this works is available from R.M.A.D. upon reasonable request.
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Acknowledgements
Parts of this research were carried out at ELBE at the HelmholtzZentrum Dresden—Rossendorfe. V., a member of the Helmholtz Association. We would like to thank U. Lehnert, J. Teichert and the rest of the ELBE team for assistance, and M. Gensch for support of the experiments. Z.W. thanks A. Renno for characterizing the samples microscopically, and Enke Liu and C. Reinhoffer for helpful discussions. S.K. and B.G. acknowledge support from the European Cluster of Advanced Laser Light Sources (EUCALL) project, which has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No 654220. N.A., S.K., and I.I. acknowledge support from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 737038 (TRANSPIRE). R.M.A.D. and P.S. were supported in part by the DFG (German Research Foundation) through the Leibniz Program and ct.qmat (EXC 2147, projectid 39085490). F.X. was supported by National Natural Science Foundation of China (Grant No. 11934005, 61322407, 11474058, 61674040 and 11874116), National Key Research and Development Program of China (Grant No. 2017YFA0303302 and 2018YFA0305601). P.v.L. and Z.W. acknowledge support by the DFG via project No. 277146847—Collaborative Research Center 1238: Control and Dynamics of Quantum Materials (Subproject No. B05).
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Z.W. and T.O. conceived the project with P.S. S.K., and Z.W. carried out the THz HHG experiments and analyzed the data with S.G., J.C.D., B.G., I.I., N.A., M.C., M.B. R.M.A.D., P.S., and T.O. performed the theoretical calculations and analyzed the data. J.L. and F.X. fabricated and characterized the highquality samples. S.G., P.v.L., and Z.W. characterized linear THz response of the samples. Z.W. wrote the manuscript with contributions from S.K., R.M.A.D., S.G., J.L., P.S., and T.O. All authors commented the manuscript.
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Kovalev, S., Dantas, R.M.A., Germanskiy, S. et al. Nonperturbative terahertz highharmonic generation in the threedimensional Dirac semimetal Cd_{3}As_{2}. Nat Commun 11, 2451 (2020). https://doi.org/10.1038/s41467020161338
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