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Abstract
Intraband high-order harmonics in solids reflect the nonlinear motion of carriers in the energy bands and have important applications. In general, multiple energy bands contribute to the intraband harmonics. We reveal the interference mechanism of intraband harmonics between electrons and holes by using the k-resolved semiclassical intraband model. The model gives a quantitative relationship between the intraband radiation yield, the energy band dispersion, and the laser parameters. Based on a preexcitation scheme, we present a method to reconstruct energy bands by using the intraband harmonics. We simulate the intraband harmonics of photo-carriers polarized by a bias field. The variation of harmonic yields as a function of a delay time results from the k-resolved energy band dispersion and reflects the Bloch oscillation of photo-carriers driven by the bias field.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent decades, with the development of ultra-strong laser technology, the study of the electron dynamic processes induced by intense lasers, especially high-order harmonic generation (HHG), grows rapidly. In particular, HHG in gases strongly promotes the advance of attosecond technologies [1–5]. Owing to the tunneling ionization [6,7] and acceleration in alternating strong fields, electrons release harmonics with a wide spectrum when they recombine with the parent nucleus [8,9]. Therefore, gas HHG has good characteristics in generating attosecond pulses [2,3] and imaging electronic orbitals [4,5]. In recent years, HHG in crystals shows advantages in generating compact and coherent extreme ultraviolet sources due to the high density [10–19]. Different from the mechanism of HHG in gases, electrons in crystals are nonlocal, showing more complex and interesting characteristics in HHG [18,19]. Previous works [11,15] show that HHG in the plateau generated by interband transitions in crystals has long and short quantum trajectories similar to those in gases, and some experimental phenomena can be explained by the interband transition mechanism [20–22]. While the theoretical work [15] suggests that the intraband mechanism generally dominates in HHG below the bandgap or using a long-wavelength pulse. The relative contribution between intra- and interband transitions should depend on both the structure of crystals and laser parameters. In particular, in Ghimire’s paper [14], a simple intraband acceleration model captures the general features of measured HHG spectra. In Luu’s paper [23], the intraband semiclassical model is in a good agreement with the experiments, and they analyze the influence of band structures on intraband harmonics. Furthermore, Lanin’s work [24] discusses the nonlinearity of intraband radiation in ZnSe and shows the possibility of reconstructing the energy bands.
If the driving field is intense and its wavelength is long enough, electrons will be reflected at the boundary of the first Brillouin zone (BZ) [16]. In a more special case, electrons move repeatedly in the BZ under the driving of an electrostatic field. This quantum phenomenon is called Bloch oscillation (BO) [25,26]. Since the uniting of ultrafast scattering [27] and the tiny size of a crystalline unit cell requires extremely strong bias fields, although similar phenomena have been observed using artificial structures [28–31], it is difficult to generate BO in crystals. However, the development of ultra-strong laser technology increases the possibility, particularly the mid-infrared and terahertz sources, which may serve as bias fields to drive electrons [32,33].
It can be seen that the semiclassical model of electron acceleration in energy bands is widely recognized in crystal HHG. However, previous studies have not given a quantitative relationship between harmonic yield and band dispersion, and the reconstruction of energy bands using intraband harmonics [24] lacks more explicit theoretical supports. Moreover, intraband harmonics are involved by multiple energy bands in general, and the interference between energy bands has not been discussed in detail. In order to clarify these issues, further quantitative descriptions are necessary. In this paper, a $k$-resolved intraband semiclassical model developed based on previous works [10–14,23] can be used to analyze the distribution of electron nonlinearity in reciprocal space. It gives the intensities and phases of intraband harmonics from different energy bands, so the interference mechanism of intraband harmonics between energy bands is revealed. The revelation of this interference mechanism is helpful to optimize the intraband harmonics. Moreover, our semiclassical model gives the quantitative relationship between intraband harmonics and band dispersion, which provides a reference theory for an all-optical mapping of the electron band structure. The dominance of the intraband harmonics in this work is verified by solving the semi-conductor Bloch equation (SBE), and the realization of the reconstruction method actually benefits from our preexcitation scheme. Besides, We propose a pump-drive-probe scheme to detect the $k$-resolved band dispersion and observe the BO generated by a bias field in crystals.
The agreement between the theory of electron intraband motion and experiments in the works [16,23] shows that electrons can maintain sufficient coherent motion driven by intense fields. Our scheme only needs to detect low-order harmonics, which reduces the experimental difficulty. We notice that, different from the interband radiation which originates from the interference of electrons in different energy bands and relies heavily on the coherence, the intraband radiation mainly originates from the nonlinear motion of electrons within the bands, which in fact does not require the coherence between electrons. So we think that the effect of the dephasing on the intraband radiation is much smaller than that of the interband radiation. Therefore, compared with the detections using the interband harmonics [34], the detection schemes based on intraband radiations are more conducive to the detection of BO and the reconstruction of energy bands.
The paper is organized as follows. In Sec. 2, we review the theoretical framework. In Sec. 3, we use the numerical results to reveal the interference of intraband harmonics from different energy bands and apply the preexcitation scheme to reconstruct the energy bands and detect BO. We conclude our work in Sec. 4.
2. Theoretical methods
We use the method in Refs. [15,35,36] to simulate the HHG process in SiO$_2$ by solving the SBE, which has been widely used in numerical simulations of crystal HHG and has been shown to be sufficient to reproduce experimental observations.
We suppose that the direction of energy bands and the polarization of the laser fields are along the $\Gamma$-$M$ direction. Because the higher conduction bands of SiO$_2$ are smoother and occupied by fewer electrons, it is reasonable to only consider the highest valence band and the lowest conduction band. The calculated energy bands are fitted by the function $\varepsilon _m(k)=\sum _{i=0}^{\infty }\alpha _{m,i}\cos (ika_0)$, with the lattice constant $a_0$=8.168 a.u. These coefficients can be found in our previous work [37]. The subscript $m=h$ and $e$ indicate valence and conduction bands, respectively.
Besides, in order to further analyze the intraband radiation, we adopt the $k$-resolved semiclassical method in our work [37], which is briefly introduced below.
Driven by an intense laser field $F$, the crystal momentum satisfies the acceleration theorem,
We simplify the driving laser into a cosine field $F(t)=-F_b\cos (\omega t)$ and according to Eq. (1), where $k_0$ is the initial momentum and $A_0=F_b/\omega$ is the peak value of the vector potential. Energy bands can be expanded by Taylor series: According to Eqs. (2) and (3), considering that the group velocity of carriers is the gradient of the energy band, i.e. $v_{m}(k)=\frac {\partial }{\partial k}\varepsilon _{m}(k)$, and using trigonometric formulas we obtain the acceleration,3. Results and discussions
3.1 Interference of intraband harmonics from electrons and holes
We adopt a pump-probe scheme similar to those in Refs. [37,38]. Experiments by Wang et al. [38] show that the pump light suppresses harmonic yields. Our simulations on SiO$_2$ in Ref. [37] by solving SBE also show similar phenomena, but the intraband harmonics dominate below the bandgap, and the decrease in harmonic yield is due to multielectron interference. Our subsequent simulations with ZnO are similar. In this paper, under the condition that the crystal is not damaged, we use a stronger pump light, so that, carriers excited by the probe light can be ignored relative to those excited by the pump light. In this case, the multielectron interference is ignored, which is conducive to the discussion of our concern.
We use a pump light resonating with the minimum bandgap energy of SiO$_2$ to preexcite enough carriers near the $\Gamma$ point in our simulations using the SBE. The wavelength of the pump light is 151.5 nm, the peak electric-field strength is $5\times 10^{-4}$ a.u., and the full width at half maximum (FWHM) of its Gaussian envelope is 70 fs. After the pump light, a Gaussian-envelope probe light with FWHM of about 100 fs and wavelength of 3500 nm irradiates the excited material. Figure 1 shows the intra- and interband harmonic yields with different dephasing times, the peak electric-field strength of the probe light $F_p$=0.005 a.u. We can see that the low-order harmonics are dominated by the intraband harmonics, when the dephasing time $T_2$ ranges from 3 fs to infinity, especially when the dephasing time is short. The laser parameters selected here ensure that the carriers contributing to the harmonics mainly come from the resonance excitation of the pump light, while the carriers excited by the probe light are very small. Note that although the pump light provides sufficient carriers, the ionization rate of electrons at each $k$ point is still less than 1 $\%$. According to our previous paper [37], due to the concentrated distribution of carriers excited by resonance, the interference between different $k$ points, especially the destructive interference, is avoided. Therefore, the preexcitation scheme can effectively enhance intraband harmonics. The interband harmonics decay rapidly with the shortening of the dephasing time, whereas, as shown in Fig. 1, the dephasing time has a slight effect on the intraband harmonics. Only when the dephasing time is shortened to 10 fs or even 3 fs, can the interference between different $k$ points affect some harmonic yields due to the broadening of electron population. In this paper, we only focus on harmonics lower than the eleventh order, and the intraband harmonics are much stronger than the interband harmonics. So, only intraband harmonics are considered and the dephasing time is set to be infinite if there is no special explanation.
Fig. 1. The HHG spectrum of SiO$_2$ simulated by solving the SBE and adopting the pump-probe scheme. The dephasing time is $\infty$ (a), 100 fs (b), 10 fs (c), and 3 fs (d), respectively. The black vertical line indicates the location of the minimum bandgap.
Simulations in the work [39] demonstrate that the $k$-space-dependent transition dipole moment has great influence on harmonics in SiO$_2$. We calculate the harmonic by using the dipole in k-p approximation and the dipole obtained by the first-principles calculations (the results are not shown here). When our preexcitation scheme is adopted, there is no obvious difference in the harmonics we care about. Because electrons are excited only from the $\Gamma$ point and the intraband harmonics are mainly related to the band dispersion, the $k$-distribution of dipole moments does not actually affect much on intraband harmonics. All the results presented are calculated using the dipole moment obtained by the first-principles calculations.
Since the preexcited carriers are concentrated around the $\Gamma$ point of the two energy bands, we exclude harmonics from more energy bands and other positions for convenience, then only consider the contribution of carriers with initial momentum $k_0$=0 within two bands in our semiclassical model. It should be noted that the coherence between excited electrons and holes is ignored when the probe light arrives in our preexcitation scheme, so the contributions of tunneling ionization and recollision to intraband harmonics [40] are neglected in practice.
After tests, for energy bands of SiO$_2$ expanded by 12-term cosine functions, 110-items in Eq. (3) for expanding energy bands are accurate enough. Taking the coefficients of Taylor series with $k_0$=0 into Eq. (5), the harmonic amplitude dependence on the vector potential peak of the probe light at $k_0$=0 is obtained. As an example, the seventh harmonic amplitudes generated by electrons $-Y_{e,7}$, holes $Y_{h,7}$, and the total of them $Y_{t,7}$ are shown in Fig. 2(a), respectively. And the seventh harmonic intensity $Y_{t,7}^2$ is shown in Fig. 2(b), for comparison with the calculation of the SBE. The red-dashed-dotted curve in Fig. 2(b) shows the seventh harmonic intensity dependence of the vector potential peak $A_0$ (corresponding to $F_b$=0 to 0.005 a.u.) simulated by solving the SBE, other parameters are the same as above. Harmonics of other orders have similar characteristics and are not shown here. The variation trend of harmonic intensity calculated by the SBE is consistent with that calculated by the semiclassical model, and the minima of harmonic intensity calculated by the SBE correspond to the positions where $Y_{t,7}$ passes through zero. The semiclassical model does not consider the envelope of the probe field, while the Gaussian envelope which is close to the pulse in experiments is used in the SBE. Considering that the peak value of vector potential is not the most representative for the Gaussian envelope field, it is reasonable that the minima of harmonic intensity calculated by the SBE deviate a little to the right. We note that the intraband harmonic amplitudes of electrons and holes at $k_0$=0 are generally the same order of magnitude when $A_0<0.35$ a.u. (carrier does not reach the boundary of the BZ), although the valence band is much flatter than the conduction band. This shows that the carriers in conduction and valence bands near the $\Gamma$ point have a similar efficiency of frequence multiplier. The change in sign of the harmonic amplitudes means that its phase changes by $\pi$. If the harmonic amplitudes of electrons and holes have the same sign, they interfere constructively, otherwise, they interfere destructively. Thus, the interference between the harmonics of electrons and holes dominates the intraband harmonic yield. Specifically, for electrons and holes, their harmonic amplitudes vary differently with the vector potential peak, and only considering their common contributions can they be consistent with the results of the SBE. This interference is common in harmonics of different orders. It can be seen that the intraband harmonic interference between energy bands is important in the HHG process involving multiple energy bands.
Fig. 2. The seventh harmonic amplitudes and intensities as a function of $A_0$. (a) The blue-solid, magenta-dashed-dotted, and green-dashed curves represent the seventh intraband harmonic amplitudes of electrons, holes, and the total of them at $k_0$=0 varying with the vector potential peak, respectively. (b) The green-dashed and red-dashed-dotted curves represent the seventh harmonic intensity calculated by the semiclassical model at $k_0$=0 and the SBE using the preexcitation scheme, respectively. The thin black curve is the square of the first term of $Y_{t,7}$.
3.2 Band reconstruction
Since the energy band is axisymmetric with respect to $k_0$=0, the odd terms in Eq. (3) are zero, and even order harmonics disappear. $Y_{t,n}$ with $k_0$=0 can be written as
When $A_0\geq \pi /a_0$, the harmonics of finite orders are sufficient to reflect the dispersion of the whole energy bands. That is, the harmonic intensities and phases can reconstruct the energy bands. This is an inverse process of obtaining $Y_{t,n}$. We square the harmonic intensity calculated by the SBE as the absolute value of $Y_{t,n}$. And the signs of $Y_{t,n}$ can be obtained by the harmonic phases and Eq. (4), here the harmonic phases are derived from the phases of the Fourier transform of the harmonics obtained by the SBE. Ignoring the contribution of higher order terms, substituting the first nine harmonic amplitudes obtained by the SBE for $Y_{t,n}$ in Eqs. (6), $c_{e,n}-c_{h,n}$ can be obtain by solving the system of linear equations, thus obtaining the band gap,
Here we use the harmonic intensities with $A_0$= 0.385 a.u. which approaches the boundary of the BZ, and the dephasing time is infinity and 3 fs, respectively. The reconstructed band gaps are shown in Fig. 3. Our calculation only takes into account the contribution of carriers excited at the $\Gamma$ point, the band gap reconstructed with the dephasing time of 10 fs slightly deviates from the band gap used in simulations, but the difference is small. Since the valence band of SiO$_2$ is very flat, we actually reconstruct the conduction band.Fig. 3. The solid-blue line represents the difference between the lowest conduction band and the highest valence band of SiO$_2$. The dashed-red line represents the bandgap reconstructed with the first, third, fifth, seventh, and ninth harmonics generated by probe lights with $A_0$=0.385 a.u. (the wavelength is 3500 nm, $F_b$=0.005 a.u., and the FWHM is 100 fs, as in Fig. 1) obtained by calculating the SBE. Because it is a Gaussian pulse, using 0.95$A_0$ here can better reconstruct the energy bands. There is no quantitative number of carriers in our analysis, the reconstructed bandgaps are multiplied by a constant and plus the minimum bandgap.
Our analysis and calculation show that it is accurate to consider only the contribution of the $\Gamma$ point, when the distribution of carriers is relatively concentrated (generally meaning that the most excited electrons have zero initial momentum, the preexcitation scheme with $T_2>$10 fs is exactly the case). Previous works (such as [24]) using only mid-infrared light to excite electrons obviously do not meet this condition. Moreover, we can see that the phases of harmonics are also nonlinear responses of the energy band structure. Not only the intensities but also the phases of harmonics are necessary for reconstructing energy bands. Furthermore, according to Eq. (6), the low-order coefficients of the energy bands do not contribute to high-order harmonics. So, it is impossible to reconstruct the energy band by using certain high-order harmonics while ignoring some low-order harmonics, even if the harmonic dependence on the driving light field strength or wavelength is obtained. We notice that the first harmonic is difficult to measure in experiments. Fortunately, the first harmonic is a necessary parameter only for determining $c_{e,2}-c_{h,2}$. This is a parabolic coefficient commonly used to calculate the effective mass and can be measured by other methods. And we can also add a constraint that the slope of the energy bands at the high symmetry points is zero (such as, $\dot {\varepsilon }_m(\pi /a_0)=0$) to determine $c_{e,2}-c_{h,2}$ when $A_0\geq \pi /a_0$. It can be seen from the example of SiO$_2$ that a few low-order harmonics can accurately reconstruct the energy bands, which reduces the experimental difficulty compared with other schemes, like interband methods. Besides, the simulations with the SBE are consistent with our analysis, that is, the effect of the dephasing on the intraband radiation is much smaller than that of the interband, which also expands the scope of the application of our scheme.
The semiclassical analysis shows that intraband harmonics sensitively depend on the band structure. It is for this reason that the band can be reconstructed using intraband harmonics. For polycrystals, the imaging methods based on bandstructure can not be applied.
3.3 Polarization of carriers and observation of Bloch oscillation
The harmonic amplitudes of carriers depend on their initial momentum. The $k$-resolved harmonic amplitudes can be obtained by expanding the energy bands at different $k_0$ points and taking the coefficients of the Taylor series into Eq. (5). Figure 4 shows the $k$-resolved harmonic amplitudes with $A_0$=0.269 a.u. The $k$-resolved harmonic amplitudes facilitate analysis of the $k$-space distribution of the electron nonlinearity. Due to the lack of symmetry, even order harmonics occur at positions other than the high symmetry points ($\Gamma$ and $M$ points). As can be seen in this figure, the amplitudes of the conduction band at the $M$ point are significantly larger than that of the $\Gamma$ point and valence band. In other words, electrons near the $M$ point of the conduction band have the strongest nonlinearity and contribute the most intraband harmonics. Carriers excited by nonresonance and intense fields have a relatively wide distribution in the $k$ space, even reaching the $M$ point. In this case, the intraband harmonics generated in the conduction band dominate.
Fig. 4. The $k$-resolved intraband harmonic amplitudes of SiO$_2$ with $A_0$=0.269 a.u. Since the harmonic amplitudes of holes [ (c) and (d)] are one order of magnitude smaller than that of electrons [ (a) and (b)], the total harmonic amplitudes which are almost the same as that of electrons are not shown here.
The $k$-resolved harmonic amplitudes can be measured by the pump-drive-probe scheme. In the numerical simulation of the SBE, we add an electrostatic field (with the strength of $F_c=5\times 10^{-5}$ a.u., polarized along the $\Gamma$-$M$ direction) to drive the carriers, and adjust the delay time $t_{delay}$ between the pump and probe lights to detect the intraband harmonics of carriers polarized at different levels. According to Eq. (1), carriers move in the $k$ space driven by the electrostatic field, and the distance of movement is proportional to the electrostatic field strength and the delay time
Under the drive of the electrostatic field, carriers move repeatedly in the first BZ. The cycle of carriers crossing one BZ corresponds to one BO period, so the BO period $T_{bo}=2\pi /F_{c}a_{0}$, about 370 fs. The FWHMs of the pump light and the probe light are 30 fs and 60 fs respectively, which are much smaller than $T_{bo}$. Other parameters of the pump light are the same as before. The wavelength of the probe light is 3500 nm, $F_b$=0.0035 a.u., corresponding to $A_0$=0.269 a.u. The delay time determines $\Delta k$, thus determining the carrier initial momentum when the probe light reaches the material. Therefore, the $k$-resolved harmonic amplitudes can be reflected by the time-resolved harmonic spectrum.The blue curves in Fig. 5 show harmonic intensities at different delay times calculated by the SBE. Photo-carriers are polarized by the electrostatic field, thus generating even-order harmonics. As revealed earlier, the intensities of intraband harmonics depend on the initial momentum of carriers. So, the harmonic intensities varied periodically with $T_{bo}$, due to the BO of carriers driven by the electrostatic field. The dashed-green lines in Fig. 5 are the $k$-resolved harmonic intensities calculated by the semiclassical method, which are mapped to one $T_{bo}$ for comparison. The variation of the time-resolved harmonic intensities is in line with the $k$-resolved harmonic intensities. Therefore, the periodic variation of harmonic intensities can reflect the BO and the $k$-distribution of the carrier nonlinearity. We consider a higher conduction band and calculate the SBE with three bands [35], the results (not shown here) are consistent with that of two bands. This means that carriers are almost all reflected at the boundary of the BZ and the probability of transition to higher bands is very small for the electric field we consider.
Fig. 5. The blue curves show harmonic intensities at different delay times calculated by the SBE. Green-dotted lines are the $k$-resolved harmonic intensities calculated by the semiclassical method, which are mapped to one $T_{bo}$ for comparison. The black vertical lines indicate the delay times for carriers to return to the $\Gamma$ point after a BO period.
4. Conclusion
In summary, we use the Taylor series to expand the energy bands and obtain the analytical relationship between $k$-resolved harmonics, energy bands, and laser parameters. When the preexcitation scheme is adopted in the process of HHG, the band structure can be accurately reflected by the carrier’s intraband harmonics. The difference in dispersion characteristics of electrons and holes along different energy bands results in different phases of intraband harmonics. When multiple energy bands provide intraband radiation, the interference of intraband harmonics between different bands affects the total harmonic yield. In addition, we propose a pump-drive-detect scheme to generate even-order harmonics and detect $k$-dependent dispersion of energy bands and BO. The $k$-resolved intraband model can be used to analyze the $k$-space distribution of intraband harmonics. As an example, we found that the intraband harmonic of the $M$ point is the strongest in SiO$_2$. This is undoubtedly of great significance for harmonic optimization. Our work sheds light on the study of intraband carrier dynamics, and we expect our theoretical work inspires people to further understand the ultrafast nonlinear dynamics of electrons in crystals.
Funding
National Natural Science Foundation of China (11674363, 91850121); National Key Research and Development Program of China (2019YFA0307702).
Disclosures
The authors declare no conflicts of interest.
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