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Tunnelling, acceleration, and collision of electrons are the basic events in the process of high harmonic generation (HHG) in strong-field interaction with atoms. However, the periodic array of atoms in semiconductor structure makes three steps become interatomic coherent process which leads to complicated carrier dynamics and two sources of high harmonic emission: interband polarization and intraband current. The difference of features of high harmonic generation between semiconductors and atoms is strongly linked to the unique presence of intraband motion which manifests itself a nontrivial role in intertwined two dynamics. Here, we review recent experimental and theoretical advances of understanding coupled interband and intraband mechanisms of HHG in semiconductors. Particularly we focus on the influence of intraband motion on the interband excitation, and on the subsequent HHG emission and attosecond pulse generation.
Advances in intense pulse lasers have opened up an avenue to field-induced nonperturbative nonlinear optical phenomena, such as high harmonic generation (HHG) and attosecond pulse generation.[1] Traditionally, the mediators of the strong-field interaction are isolated atoms or small molecules.[2–7] Recently, a range of semiconductors have been used to produce high harmonics when driven by visible,[8] mid-infrared,[9,10] and terahertz[11] fields, allowing for a new spectroscopic technology of reconstructing the electronic band structure[12] and generating pulses with duration of a few tens of attoseconds.[13,14]
Semi-classical three-step model of electrons in atoms offers the fundamental insights into microscopic HHG process.[2] The model comprises tunneling through the bending potential barrier, acceleration of electron in free space and recombination with its associate core. However, the HHG process in semiconductors is much more complicated due to the dense array of atoms in the system.[15–18] The comparison of microscopic mechanisms for atomic and solid-state HHG in both real and momentum space is depicted in Fig.
![]() | Fig. 1. (a)–(c) The real-space picture of solid-state and atomic HHG. In (c) three-step recollision model of atomic HHG includes tunelling, acceleration and recombination whereas in (a) tunnelled electrons scattered by periodic Coulomb potential and they could collide with not only its parent ion but also neighboring ions. (b)–(d) The picture of solid-state and atomic HHG in momentum space. In panel (d) tunneled electron accelerates freely in continuum state and recombines with its parent ion whereas in panel (b) two-band model of solid-state HHG where both interband polarization and intraband current could emit high-harmonic radiation. Adapted from Ref. [19]. |
The first experiment of HHG was observed in bulk ZnO crystal[9] exposed to mid-infrared laser pulse, marking the breakthrough of research in HHG from semiconductors. The observed harmonics spectrum extends well beyond the band edge of the crystal which manifests the significant effect of Bloch oscillation to the nonlinearities of radiation and validates the linear scaling of the harmonic cut-off energy to the field amplitude. Besides, the experiment of terahertz-induced HHG from GaSe[11,25] demonstrates that dynamical Bloch oscillation contributes considerably to the HHG process and leads to the appearance of second plateau in the HHG emission spectrum. Bloch frequency,
In this Review, we begin with the theoretical methods of describing and analysing HHG from semiconductors. Then we present an overview of recent progress of understanding intraband motion effect on the behavior of interband excitation in the coherent microscopic process and outcome of harmonic emission. Lastly we summarize works with regard to difference of HHG process between semiconductors and atoms.
For the optically excited materials description, the relevant physical quantities are the microscopic polarization
The two-band SBEs read
Analyzing a temporal or spatial signal in frequency domain to gain more properties is applied extensively in the field of image processing. Here we review the development of time–frequency methods. Fourier transform (FT) is the basic algorithm and its integration goes over the whole time domain and the resulting signal contains the statistic property of frequency, one cannot know the exact moment of generating a signal with any frequency. To overcome the drawback of FT, new algorithms based on the idea of short time FT have been proposed since 1940s, like Gabor transform (GT)[31] and wavelet transform (WT).[32] For a temporal signal f(t), GT is defined by
However, the uncertainty relation of time and frequency characteristics is inherent to these transforms. The synchrosqueezing transform (SST)[33] is proposed to lower the uncertainty greatly. SST is a kind of post-processing method by reassigning the coefficients in frequency[34] and is originally introduced in the context of continuous wavelet transform (CWT),[35] which is defined by
CWT-based SST is firstly applied in strong-field physics to analyze near- and below-threshold high-order harmonics of atoms which validates the possible semiclassical analysis and is better to understand working mechanisms for below-threshold generations.[36] Although SST is a powerful tool in analyzing instantaneous properties, due to the wave shape functions used in CWT is band-limited and the low resolution in high frequency part, CWT-based SST would not be suitable to address the problem in the circumstances requiring high resolution in low and high frequency part of time–frequency analysis at the same time. Synchrosqueezing with the short-time Fourier transform (SSTFT)[37] has been proposed to address this problem but it is not suitable to estimate instantaneous high frequency accurately. All problems above are overcame by synchrosqueezed wave packet transform (SSWPT)[37,38] which is capable of estimating a wide band of instantaneous frequency accurately and has a better resolution to distinguish high frequency harmonic than the CWT-based SST. A geometric parameter s is used to define a family of wave packets
Given that more adjustable parameters in SSWPT one is encouraged to apply it in strong field physics, particularly HHG, to get a detailed and accurate information of radiation distribution in time–frequency plane.
In the previous investigations of resonant nonperturbative extreme nonlinear optics, it is usually thought that the interband transition near the band gap in semiconductors to a great extent decides the nonlinear optical response, such as carrier-wave Rabi flopping (CWRF), and the intraband dynamics can be neglected in theoretical analysis.[39,40] However, the recent studies have shown that the intraband current is strong and leads to a occupation distribution of conduction band in a large fraction of Brillouin zone even in the CWRF regime where
Since the interband transition rate decreases exponentially with increasing energy gap, the analysis of population dynamics initially excited from
![]() | Fig. 2. (a) Time-resolved electronic population in the lowest conduction for selected reciprocal-space pathways. The starting point is k = 0 for the blue curve and ![]() ![]() |
In Ref. [42], it was further demonstrated that the intraband motion can dramatically affect the interband excitation. As stated in the abstract of the paper, “Surprisingly, we found that despite the resonant driving laser, the optical response during the light–matter interaction is dominated by intraband motion.” It is manifested that the HHG in ZnO crystal induced by pump pulse with similar parameters of that in Ref. [41], the analogous behavior for carrier dynamics of k-integrated conduction band occupation (as displayed in Fig.
All these studies demonstrate that the intraband motion plays an non-negligible role in carriers transition among bands which is quiet different from atomic case. According to the semi-classical analysis in atoms, the ionization takes place at k = 0 and the transition rate is only sensitive to the external field, the weight of a classical trajectory is fixed during the free acceleration in the continuum state.[43–46] In the HHG process of semiconductors, the instantaneous population of electron–hole pair is sensitive to the field strength and energy band during the intraband motion.
Recently, a strong CEP dependence of far-from-resonantly excited HHG from wide-gap MgO crystal has been observed experimentally[47] (see Fig.
![]() | Fig. 3. (a) Experimentally measured CEP dependence of HHG emission from MgO crystal. Adapted from Ref. [47]. (b) Simulated results of panel (a). (c) Same computation of (b) but in the absence of intraband motion. Adapted from Ref. [51]. |
However, recent investigation[51] shows that even in this case, the intraband motion plays a crucial role on the CEP dependence of HHG in MgO crystal. It is illustrated that in the calculation of HHG with field amplitude
![]() | Fig. 4. (a) The total HHG spectrum ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
It is demonstrated that the temporal confinement of interband excitation by intraband motion still exists for longer duration (see Figs.
It is counterintuitive to find that in contrast to the case of pure interband mechanism, the population of conduction band for coupled process is lowered at the end of pulse due to the CWPT effect, but leads to the higher efficiency of harmonic emission. The temporal confinement of interband excitation within each half cycle facilitates the constructive interference among the quantum carriers in the transition process and consequently enhance the nonlinear interband polarization and makes it the dominant contribution to the total HHG.[51] The influence of intraband motion on the interband excitation is universal and intensively influence the behavior of high harmonics emission.
It is revealed that the temporal confinement of interband excitation happens roughly upon the moments when the vector potential equals to zero at the crest of electric field.[51] For HHG induced by a tailored two-color pulse like
![]() | Fig. 5. (a) HHG spectrum for the two-color field with relative phase ![]() ![]() |
The deviation of time profile of solid-state HHG process from atomic case embodies the spatial delocalization of Bloch electrons in the strong-field limit.[52,53] In Ref. [52], the HHG yield exhibits anisotropic angular distribution for a linear polarized pump in MgO crystal. The underlying origin is assumed to be coherent collisions of Bloch electrons with neighboring atomic sites in periodic solids and is qualitatively validated based on the simple semi-classical electron trajectory analysis. This is further justified by the strong laser ellipticity dependence of HHG. It is found that the highest and secondary highest HHG efficiency occur when the trajectories are driven to traverse across the Mg–O and Mg–Mg (or O–O) sites with large electronegativity gradient for charge transfer along these bonding directions. The minimum happens when the electron trajectories are collision-free. The period for trajectories connecting Mg–O sites along the X–
We review the recent investigations of coupled interband polarization and intraband motion dynamics in solid-state HHG. The reported critical temporal confinement of interband excitation effect is common in semiconductors regardless of excitation parameters. The effect intuitively interprets the domination of interband polarization in the emission owing to the intraband motion and can greatly enhance extreme nonlinear optical effects. The interband excitation is sensitive to both electric field and vector potential condition and the emission is 1/4 optical cycle delayed which is related to the delocalization process in real space. The unusual temporal structure of interband excitation and harmonic pulse generation in solid allows controlling of ultrafast carrier dynamics on the level of carrier wave by tuning the waveform of multi-color field. For the tailored two-color field with
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