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Abstract
We investigate the electronic excitation of solids in strong fields by solving the time-dependent Schrödinger equation. The excitation probability exhibits a strong modulation as a function of laser intensity when the initial states fill in the whole valence band. To have a clear insight into the modulation, we further study the electronic excitation from a single eigenstate in solids. A series of resonance-like enhancements of excitation probability are produced by changing the laser intensity and wavelength. We attribute the resonance-like enhancements to the channel-closing effects in solids. It is shown that the excitation probability exhibits enhancements when the value of channel is odd for intracycle interference and an integer for intercycle interference. This is different from the atom that the enhancement occur in the integer channels. We also reveal that the channel-closing effects can be observed by solid high-order harmonic generation.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the rapid development of laser technology, numerous interesting strong-field phenomena have been observed [1–5]. Above-threshold ionization (ATI) and high-order harmonic generation (HHG) in the atom as the prominent processes among these phenomena have been extensively studied over the past decades [6–18]. A very intriguing feature of ATI or HHG lies in the dependence of these phenomena on the intensity of the laser field [19–22]. The photoelectron or high-order harmonic present resonance-like enhancements as a function of the laser intensity, such that an alteration of a few percent in the field strength may enhance the spectral intensity by an order of magnitude. Previous investigations [23–25] show that the resonance-like enhancements are attributed to the closing of a multiphoton ionization channel in a laser field. The ionization peaks when
is interger, where $R$ indicates the number of photons absorbed by the atom and is called the channel-closing number. $I_p$ is the ionization potential of the target atom, and $\textit {U}_p=F_0^2/4\omega ^2$ is the ponderomotive energy. $F_0$ and $\omega$ are the amplitude and angular frequency of the laser field, respectively. The enhancements merely appear in a portion of the ATI spectra (located at $6U_p$ and $8U_p$) and the harmonic spectra [25]. For ATI, the low-energy part of the spectrum and the part preceding the cutoff depend much more smoothly on the intensity.Compared with the atom, the solids exhibit unique optical and electronical properties with periodic structure and high density. Recently, the study of strong-field electronic dynamics in bulk solids has attracted extensive attention [26–42] due to its potential impact on future technologies such as emerging petahertz electronic signal processing [43,44] or strong-field optoelectronics [45,46]. Excitation probability induced by a laser field as a crucial step plays a fundamental role in electronic dynamics in solids. It is important to understand the underlying mechanism of the electronic excitation.
In this work, the electronic excitation from valence band (VB) to conduction bands (CBs) in strong fields is investigated by solving the time-dependent Schrödinger equation. The excitation probability is maximized at specific intensity when the initial state fills in the whole VB. We further study the electronic excitation in solids from a single eigenstate. A series of resonance-like enhancements of excitation probability are produced with specific laser intensities and wavelengths. We attribute the resonance-like enhancements to the channel-closing effects in solids. However, different from the atom, the enhancements occur when the value of channel is odd for intracycle interference and an integer for intercycle interference. The resonance-like enhancements of excitation probability also exist in other eigenstates. We also propose a scheme to observe the channel-closing effects using solid HHG.
2. Theoretical model
2.1 Electronic excitation in solids
To investigate the electronic excitation in solids, we numerically solve the time-dependent Schrödinger equation (TDSE) based on a one-dimensional periodic structure. In the length gauge with the dipole approximation, the time-dependence Hamiltonian is written as
where $\hat H_{0}=\hat p^2/2+v(x)$ is the field-free Hamiltonian and $F(t)$ is the electric field. $v(x)$ is the periodic potential of the lattice. Atomic units are used unless stated otherwise. Herein, we use the Mathieu-type potential $v(x)=-v_0[1+\cos (2\pi x/a_0)]$ to describe the lattice potential of solids, with $v_0=0.37$ a.u. and lattice constant $a_0=8$ a.u. The parameters are chosen to mimic the structure of a semiconductor with wide band gap [47–50], such as the Zincblende AlN. The Mathieu-type potential [51] has been widely used in optical lattice research area [52,53] and strong-field electronic dynamics in solids [54–59]. We perform all calculations in the coordinate space within the region $[-396, 396]$ a.u. ($99$ lattice periods). To overcome the unphysical reflections of the wave function at the edges of the grid, an absorbing boundary $g(x)$ is used and it is given byIn the absence of an external laser field, the eigenvalue equation of the field-free Hamiltonian $\hat H_0$ is written as
where $n$ is the eigenvalue number and $\varphi _n(x)$ is the corresponding eigenstate. The energy bands can be obtained by solving the eigenvalues and eigenstates of $\hat H_0$ on the coordinate grid. The obtained band structure calculated with the diagonalization scheme is shown in Fig. 1(a). The VB and the first conduction band (CB1) can be clearly distinguished. The eigenstate numbers $100-198$, $199-297$ correspond to VB and CB1, respectively. The energy gap between VB and CB1 is $E_g=4.2$ eV. The other bands are also calculated but not shown here. We also calculate the band structure by using the Bloch-state basis [56,57]. The obtained band structure is shown in Fig. 1(b). Comparing the energy bands in Figs. 1(a) and 1(b), one can see that the number of bands, the width of the bands and the band gaps obtained by the two methods are in good agreement. This confirms the accuracy of our calculated field-free energy bands.
Fig. 1. (a) The band structure calculated by the diagonalization scheme in coordinate space. (b) The band structure calculated by Bloch-state expansion in reciprocal space. Only two bands are shown.
With the single-active electron approximation, the motions of electron of different eigenstates can be solved separately. The evolution of the time-dependent wave function $\psi (t)$ can be obtained by numerically solving the TDSE with the second-order split-operator method [60]. In order to make the electric field have the same form in every cycle, the laser pulse is described by $F_0\cos (\omega t)$, where $F_0$ is the amplitude. The total duration of this laser pulse is four optical cycles. We also perform the electric field with a trapezoidal envelope to simulate excitation probability. The total durations is six and ten cycles, respectively. Comparing the excitation probability calculated with and without the envelope, they have the same resonant features.
After the laser pulse is gone, a fractional electrons survive at CBs. The population of the electrons remaining on CBs is called real population [44]. To obtain the real population, we project the wave function onto the eigenstates on CBs
The movement of electrons in the laser field produces a current. The laser-induced current $j(t)$ can be obtained from the time-dependent wave function:
The harmonic spectrum is obtained from the Fourier transform of the laser-induced current:2.2 Ionization in the atom
We numerically solve TDSE in length gauge to obtain the ionization in the atom. The one-dimensional model potential of the atom is given by
where $a=0.7242$ to obtain the ionization potential that close to helium $I_p=0.9043$. The ground state wave function is obtained using imaginary time propagation method. We perform the calculation in the coordinate space within the region $[-400, 400]$ a.u. To avoid spurious reflections from the spatial boundaries, the electron wave function is multiplied by an absorbing boundary. We obtain the ionization probability of the atom by counting the absorbed wave function. The laser pulse is described by $F_0'f(t)sin(\omega ' t)$. $f (t)$ is the trapezoidal envelop with $2$ cycle rising and $2$ cycle falling edges and $3$ cycle plateau.3. Result and discussion
Figure 2(a) shows the ionization of the atom as a function of laser intensity. The laser intensity changes from $530$ to $850$ $\textrm{TW/cm}^{2}$ and the wavelength is fixed at $550$ nm. It is shown that the ionization peaks at specific intensities. These peaks are located at integer channels (the upper axis). We also show the total excitation probability (the sum over excitation probabilities of all eigenstates) of solids in Fig. 2(b). The intensity varies from $0.3$ to $2.2$ $\textrm{TW/cm}^{2}$ and the wavelength is fixed at $2200$ nm. Compared with the laser intensity used in the experiment of ZnO bulk crystal [28], the intensity used in this paper will not damage the material. As shown in Fig. 2(b), the excitation probability also exhibits a strong intensity-dependent modulation. The interval of modulation is about $0.4$ $\textrm{TW/cm}^{2}$.
Fig. 2. (a) The ionization probability of the atom as a function of the laser intensity. The intensity varies from $530$ $\textrm{TW/cm}^{2}$ to $850$ $\textrm{TW/cm}^{2}$. The wavelength is fixed at $550$ nm. (b) The sum over excitation probability of all eigenstates of solids. The intensity varies from $0.3$ $\textrm{TW/cm}^{2}$ to $2.2$ $\textrm{TW/cm}^{2}$ and the wavelength is fixed at $2200$ nm.
To have a deeper insight into the intensity-dependent modulation, we select the eigenstate on the top of VB as the initial state to investigate the electronic excitation. The result is presented in Fig. 3(a). Similar to the atom, a series of resonance-like enhancements are produced at specific laser intensities. Moreover, some small oscillations appear among the enhancements and the number of oscillations between adjacent enhancements is constant. We also simulate the excitation probability driven by laser pulses with different intensities ($0.01$ $\textrm{TW/cm}^{2}$ $\leq$ $I_0$ $\leq$ $2$ $\textrm{TW/cm}^{2}$) and wavelengths ($1500$ nm $\leq$ $\lambda$ $\leq$ $2300$ nm). The result is shown in Fig. 3(b). One can see that the excitation probability exhibit regular shaped ridges and the shaped ridges occur more frequently with the increase of intensity or wavelength. As the laser intensity decreases, the shaped ridges disappear gradually.
Fig. 3. The eigenstate on the top of VB is selected as the initial state. (a) The excitation probability of electrons driven by the laser plus with the intensity varies from $0.3$ $\textrm{TW/cm}^{2}$ to $2.2$ $\textrm{TW/cm}^{2}$ and the wavelength is fixed at $2200$ nm. (b) The excitation probability driven by laser pulses with intensities from $0.01$ to $2$ $\textrm{TW/cm}^{2}$ and wavelengths from $1500$ to $2300$ nm.
3.1 Mechanism of electronic excitation in solids
To understand the underlying mechanism of the shaped ridges, we investigate the electronic excitation processes based on an ensemble of electronic trajectories in the reciprocal space. The trajectory ensemble is formed by the trajectories from different excitation times $t_i$ and different initial crystal momentum $k_0$. In this case, the electron can be excited from the VB to the CB1 with certain probability. When the depletion of the initial state can be neglected, the complex amplitude of excitation rate can be written as [30,37]:
$S_{v,{k}_0}(t_i,t_0)=\int _{t_0}^{t_i}d\tau E_v({k}(\tau ))$ and $S_{c,{k}_0}(t_i,t_0)=\int _{t_0}^{t_i}d\tau E_c({k}(\tau ))$ are the phases gathered in VB and CB1, respectively. $\Omega _{cv}(t)=-i{d}_{cv}({k}(t)){F}(t)$ is Rabi frequency and ${d}_{cv}({k}(t))$ denotes the transition matrix elements between VB and CB1, where ${k}(t)={k}_0+A(t)$ and $A(t)$ is the vector potential of the laser field. In our simulation, the eigenstate at the top of the VB is chosen as the initial state, which indicates ${k}_0=0$. This method fully includes quantum interference effects, thus the trajectories of different excitation times are coherent with each other.Based on the trajectory ensemble, the interference among the electronic trajectories within different excitation times is investigated. The intensity of the interference is given by $\left |\int _{t}^{t'}d\tau \chi (\tau )\right |^2$, which represents the probability of electronic excitation. We first consider the trajectories excited in two adjacent optical cycles. The two cycles can be divided into $\tau _1$, $\tau _2$, $\tau _3$ and $\tau _4$, as shown in Fig. 4. $\tau _1$ and $\tau _2$ belong to the same cycle, and the interference of the trajectories within $\tau _1$ and $\tau _2$ is called intracycle interference. $\tau _1$ and $\tau _3$ belong to different cycles, and the interference of the trajectories within $\tau _1$ and $\tau _3$ is called intercycle interference. The intra- (the solids line) and intercycle interference (the dashed line) with wavelength from $1700$ to $2500$ nm are shown in Fig. 5(a), respectively. Laser intensity is fixed at $1.4$ $\textrm{TW/cm}^{2}$. The peaks of intracycle interference appear at particular wavelengths $1830$ nm, $2024$ nm, etc. The peaks of the intercycle interference appear at $1830$ nm, $1928$ nm, $2024$ nm, etc. One can see that the wavelength difference between two adjacent intercycle interference peaks is almost half of that for the intracycle interference. We superimpose the complex amplitudes of the trajectories excited in $\tau _1$, $\tau _2$ and $\tau _3$. The module (the dotted line) is shown in Fig. 5(a). It is shown that the peaks appear at the wavelength $1830$ nm, $2024$ nm, etc, where the intra- and intercycle interference reach a maximum. The secondary peaks occur at $1724$ nm, $1928$ nm, etc. At these wavelengths, the intracycle interference reaches a minimum and the intercycle interference reaches a maximum.
Fig. 5. The eigenstate on the top of VB is selected as the initial state. (a) The normalized intra- (the solids line) and intercycle interference (the dashed line) with the driving wavelength $\lambda$ from $1700$ to $2500$ nm. Laser intensity is fixed at $1.4$ $\textrm{TW/cm}^{2}$. The interference of the trajectories within $\tau _1$, $\tau _2$ and $\tau _3$ (the dotted line) is also shown. (b) The normalized interference of the trajectories within 2, 3 and 4 cycles, respectively.
In order to simplify the analysis of intra- and intercycle interference, we consider several representative trajectories. Since the excitation probability is maximized when the electron passes near $k=0$. When eigenstate $k_0=0$ is selected as the initial state, we select $t_1=T_0$, $t_2=\frac {3T_0}{2}$, $t_3=2T_0$ as the excitation times of the trajectories, respectively. The complex amplitudes are expressed as
Then we discuss about the small oscillations of the excitation probability. Figure 5(b) shows interference of the electronic trajectories within $2$, $3$ and $4$ cycles, respectively. One can see that the number of the oscillations between adjacent enhancements changes from $2$ to $6$ with the increase of optical cycle. When one more optical cycle is involved, two more small oscillations appear. This is similar to that of multiple slit interference.
The excitation of the eigenstates from $k_0=-0.1214$ to $0$ and corresponding values of $R$ are shown in Fig. 6. Note that excitation of the eigenstates from $k_0=0$ to $0.1214$ is symmetric with the result shown in Fig. 6. The driving wavelength $\lambda$ varies from $1700$ to $2500$ nm and the laser intensity is fixed at $1.4$ $\textrm{TW/cm}^{2}$. It is shown that the enhancements of excitation occur at specific eigenstates and wavelengths. The dotted and dashed lines represent the integer and odd channels of inter- and intracycle interference, respectively. One can see that the two channels will shift as the eigenstate changes. When the dotted line intersects the dashed line, i.e, $R$ is odd for intracycle interference and an integer for intercycle interference, the excitation will exhibit enhancements.
Fig. 6. The electronic excitation of the eigenstates that from $k_0=-0.121$ to $0$. The driving wavelength $\lambda$ varies from $1700$ to $2500$ nm and the laser intensity is fixed at $1.4$ $\textrm{TW/cm}^{2}$. The dotted lines represent the integer channel closing numbers of intercycle interference from $11$ to $18$. The dashed lines represent the odd channel closing numbers of intracycle interference from $9$ to $17$.
The above results and analysis show that the electronic excitation in solids is very different from that in the atom. This difference is attributed to the band structure of solids system. In solids, the electrons excited at different moments have the same finial momentum when the laser pulse is over. Therefore, intra- and intercycle interference can happen among the electronic trajectories. As we discussed above, the excitation probability exhibits enhancements when $R$ is odd for intracycle interference and an integer for intercycle interference. In the atom, irradiated by laser field, the electrons are localized in the ground state and have opportunities to tunnel to continue. The electrons that are ionized with half-cycle-time interval, e.g., ionization at $t=T_0$ and $\frac {3T_0}{2}$ (shown in Fig. 4), experience reversed electric field. Thus the momentums of the electrons are opposite. The electronic trajectories do not interfere with each other. However, for the electrons ionized with one-cycle-time interval, e.g., ionization at $t=T_0$ and $2T_0$, the momentums of the electrons are the same. The electronic trajectories can interfere with each other. As a result, the ionization probability of the atom peaks in integer channels.
3.2 Observation of the channel-closing effect in solids
We next consider how to observe the channel-closing effects in solids. One possible method is using the photoemission spectroscopy. Here we propose an all-optical method based on the HHG in solids. Figure 7 shows the harmonic yield and the corresponding excitation probability as a function of $R$, respectively. The initial state is the eigenstate on the top of VB. Same as the excitation probability, one can see that the harmonic yield is also drastically enhanced in the odd channels. It indicates that the harmonic yield and the excitation probability have a strong relevance and it is feasible to observe the channel-closing effects by detecting HHG in solids. We also simulate the excitation probability and the harmonic yield when all the eigenstates on VB are selected as the initial state. The result shows that the harmonic yield and the excitation probability also have a relevance.
Fig. 7. The normalized harmonic yield and the real population on CB1 with $R$ from $9.8$ to $18.6$. The eigenstate on the top of VB is selected as the initial state. The wavelength changes from $1700$ nm to $2500$ nm and the intensity is fixed at $1.4$ $\textrm{TW/cm}^{2}$.
Figure 8 shows the harmonic spectra with $R=14.6$, $15$ and $15.8$ , respective. The eigenstate on the top of VB is selected as the initial state. The plateau (from 5 to 19 orders) of the harmonic spectrum typically originates from the transition of the electron from CB1 to VB. It is shown that the harmonic intensity is enhanced throughout the plateau in odd channels. In contrast, the enhancement of HHG in the atom only occurs in a portion of the plateau. When solids are irradiated by a laser pulse, electrons in VB have opportunities to excite into CB1. The exciting electrons are not fixed at the crystal momentum where they are excited. The Bloch electrons will be accelerated with the laser field following $\hbar d{k}/dt=-e{F}$, and can move a wide range of $k$. As a result, the electrons can transit back from CB1 to VB at different $k$, emitting photons with energy covering the whole plateau regions. This explains why the channel-closing effects contribute to the enhancement throughout the plateau region in solids.
Fig. 8. The harmonic spectra with $R=14.6$ (upward-pointing triangles), $15$ (circles) and $15.8$ (downward-pointing triangles). The eigenstate on the top of VB is selected as the initial state.
4. Conclusion
We investigate the electronic excitation in solids by solving TDSE in length gauge. With the initial state filling CB, the excitation probability exhibits obvious intensity dependent modulation. We also select an eigenstate as the initial state. The excitation probability exhibits intense intensity and wavelength dependences demonstrating resonance-like enhancements and small oscillations. These phenomena are explained by the interference of the electronic trajectories with different excitation times. We further relate the modulation to the channel-closing effects in solids. The results show that the enhancements only occur when the value of channel is odd for intracycle interference and an integer for intercycle interference, which is different from that in the atom. We also confirm that the channel-closing effects can be detected by solid HHG.
Funding
National Key R&D Program (2017YFE0116600); National Natural Science Foundation of China (91950202, 11627809, 11874165, 11704137, 11774109); Science and Technology Planning Project of Guangdong Province (2018B090944001).
Acknowledgments
Numerical simulations presented in this paper were carried out using the High Performance Computing Center experimental testbed in SCTS/CGCL (see http://grid.hust.edu.cn/hpcc).
Disclosures
The authors declare no conflicts of interest
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