1. Introduction

Carrier injection from the valence to the conduction band in semiconductors forms the basis of modern ultrafast optoelectronics. Recently, it has been demonstrated that strong-field carrier injection via quantum tunneling allows electric currents to be manipulated by the instantaneous light field, therefore opening the way to extending electronic signal processing into the petahertz ($10^{15}$ Hertz) domain [1,2]. Quantum tunneling is a highly nonperturbative process. More than fifty years ago, Keldysh introduced a dimensionless parameter in his seminal work, now known as Keldysh parameter $\gamma$, which classifies strong-field ionization into two limits: multiphoton ionization with $\gamma \gg {1}$ and tunneling ionization with $\gamma \ll {1}$ [3]. The Keldysh theory applies not only to atomic gases but also to solids where the ionization process corresponds to the transfer of an electron from the valence band into the conduction band. In solids, the Keldysh parameter is defined as $\gamma =\frac {1}{2}\sqrt {\frac {I_p}{U_p}}$ which differs from the expression for atomic gases by a factor $1/\sqrt {2}$. Here, $I_p$ is the band gap, i.e., the smallest energy difference between the conduction and valence bands in $k$-space, and $U_p=\frac {E^2}{4\omega ^2}$ is the ponderomotive energy. The regime where $\gamma \sim {1}$ is usually denoted as nonadiabatic ionization. Over the last two decades, nonadiabatic ionization in gases has been investigated extensively [4–8]. In this regime, the cycle-averaged quasistatic approximation fails as the subcycle ionization dynamics plays an important role. It has been demonstrated that the electron can be excited and accumulate energy during the propagation below the barrier, leading to a nonzero initial longitudinal velocity and a reduced tunneling exit point as compared with the adiabatic picture [9]. But whether or to which extend the subcycle-resolved nonadiabatic tunneling dynamics picture built from gases can be directly extended to solids is still unclear.

The ionization dynamics in solids when driven by a strong laser pulse is considerably more complex than in atomic systems. In addition to interband transitions from the valence band to the conduction band, in solids the laser field can also accelerate electrons within one specific band (intraband acceleration) [10]. Their relative strength and their interplay in solid-state high-order harmonic generation (HHG) have been highly discussed and debated since the first HHG experiments in ZnO [11–15]. Especially, it has been found that intraband acceleration can affect strongly the electron injection from the valence into the conduction band [16,17]. Recently, it has been proposed that an additional pre-acceleration step prior to the ionization can be relevant [18]. This is in contrast to the usual assumption in semiclassical three-step model for HHG in solids, where the electron is initially located at the top of the valence band and promoted vertically, i.e., at the same crystal momentum, into the conduction band [12,19,20]. In the semiclassical analysis developed in Refs. [12,19,20], the classical approximation is adopted by setting the band gap $E_g=0$ when solving the saddle-point equation, which allows for a real valued solution. Adopting the classical approximation means that the dynamics during the tunneling process is disregarded. Subsequent investigations indicate the importance of including the dynamics during tunneling [21–25]. In Refs. [22,24], the saddle point equations are solved in the complex plane but under the assumption that the imaginary part of the saddle point is very small. With this approximation, the saddle point equation can be solved analytically. As stated in [24], this applies to situations when the trajectories start near the extrema of the laser field and is only valid in the adiabatic tunneling regime. The nonadiabatic tunneling in solids, however, has not yet been fully understood so far.

In this paper, we investigate the tunneling dynamics in solids for different Keldysh parameters ranging from the adiabatic to the nonadiabatic tunneling regime using a complex saddle-point approach without the above mention approximations [26,27]. The complex saddle-point method has been successfully applied to analyze strong-field photoionization and HHG in atomic and molecular systems [4,28–31] and recently has been extended to solids [23,25]. With this method, we first confirm the pre-acceleration model and then find that for adiabatic tunneling, electrons with different initial momenta are accelerated to the $\Gamma$ point where tunneling from valence band to the conduction band occurs. However, in the nonadiabatic tunneling regime, the acceleration process is shortened, and tunneling can occur already before the electron reaches the $\Gamma$ point. Therefore, consistent with the atomic case, in this case the electron has a nonzero initial velocity and a reduced tunneling distance as compared to the quasi-static tunneling model.

2. Theoretical approach

Within a two-band model and following the Keldysh theory, the wave function of the conduction and valence bands can be expressed in the form of the Volkov-type wave function [3,32]

Here, $u_{ {m},{\bf {k}}}$ is the periodic part of the Bloch function, with $ {m=c, v}$ representing the conduction and valence band, respectively. $\varepsilon _{ {m}}[{\bf {k}}(t)]$ is the band energy $ {m}$. ${\bf {k}}({t})={\bf {k_{\rm{0}}}+{\bf {A}}}({t})$ where $\bf {k_{\rm{0}}}$ is the crystal quasimomentum and ${\bf {A}}({t})=-\int \bf {E}({\tau })\mathrm {d}{\tau }$ is the vector potential of the laser field. Beyond the quasistatic approximation, the laser field can have an arbitrary waveform. The probability amplitude of the electron transition from the valence band to the conduction band can be written as [32]

The above integral can be evaluated by a saddle point method which has successfully been applied to provide a clear physical picture of HHG and photoionization in atoms and molecules based on complex trajectories (also called quantum path), and recently has been extended to the solid-state HHG [23]. Here we focus on applying the saddle point method to photoionization in solids. With the saddle point method, the integral over time can be represented as a sum of contributions from all relevant saddle points $t_{s}$ before $t$ [27]

The summation goes over the integer $s=0, 1,\ldots \, , \rm{max}$, where $t_s< t_{s+1}<t_{\rm{max}}<t$. $ {s}=\sqrt {2\pi /i {w}^{\prime \prime }( {t}_s)}g(t_s)$ is the amplitude part of the saddle point $t_s$, where $w^{\prime \prime }(t_s)=(\varepsilon _{ {g}}[\bf {k}])^{\prime }$ is the second-order derivatives of exponential terms in saddle-point equations and $g(t_s)={\bf {d}}_{{cv}}[{\bf {k}}({t}_ {s})] \cdot {\bf {E}}({t}_{s})$.

The saddle points $t_s$ are obtained by solving the saddle equation

Semiconductors and dielectrics have a finite bandgap, which means the saddle point equation Eq. (4) has only complex solutions $t_s = t^r_s + i\cdot t^i_s$ [23,25], the hallmarks of quantum effects. Each saddle point corresponds to a quantum path. Transferring the saddle method developed for atomic gases to solids, the complex time $t_s$ can be identified as the moment when the electron begins to enter the potential barrier, see Fig. 1(a) [27]. The real part of the saddle point $t^r_s$ corresponds to the time when the electron exits the potential barrier. The time evolution of the quantum path under the barrier in the complex-time plane is set parallel to the imaginary-time axis, i.e., keeping the real part constant, on the basis of the experimental evidence that the measured tunneling delay time is virtually zero. When the path hits the real axis, i.e., at $t^i_s=0$, the electron emerges out of the potential barrier. After that it evolves along the real axis and all dynamical variables are real. The imaginary part of the action in Eq. (4) is associated with the modulus of the ionization probability amplitude. This imaginary part is only accumulated during the evolution along the imaginary time axis between $t_s$ to $t^r_s$. This means that the ionization probability amplitude of quantum path changes only during the tunneling. Therefore, the imaginary part of the saddle point $t^i_s$ defines the ionization probability. In momentum space, we can obtain the instantaneous momentum at the moment of tunneling ionization via ${\bf {k}}({{t}_{{s}}})={\bf {k}}_{\rm{0}}+{\bf {A}}({{t}^{{r}}_{{s}}})$ for a given crystal quasimomentum $\bf {k}_{\rm{0}}$. Correspondingly, the initial velocity $v_0$ of electron when emerging in the conduction band is

 

Fig. 1. Schematical illustration of the electron ionization in semiconductors. (a) Saddle point method in a solid. $t_s$ is identified as the moment when the electron begin to enter the potential barrier. The real part of the saddle point $t^r_s$ corresponds to the time when the electron exits the potential barrier. The imaginary part of the saddle point $t^i_s$ defines the ionization probability. (b) and (c) show schematically the electron motion and ionization in momentum space for the adiabatic and the non-adiabatic cases, respectively.

Download Full Size | PDF

In the following, we demonstrate that consistent with the atomic response, the initial velocity of an electron at the tunneling exit can indeed be nonzero in the nonadiabatic tunneling regime and reduces to approximately 0 in the adiabatic limit. The instantaneous momentum distribution at the moment of tunneling ionization depends sensitive on the laser intensity and wavelength. We take a 1-D $\rm{\alpha -quartz}$ structure as an example. The band structure is obtained by first principles density-functional-theory software (Quantum Espresso is used [33]). For convenience, we expand the obtained energy band by a Fourier expansion as [20,34]:

With $ {m}= {c,v}$ for the conduction and the valence band, respectively, $D=\rm{GM, GK}$ denoting the different lattice directions and the lattice constant $ {a}_{\rm{0}}^{\rm{D}}$. All parameters are listed in Table 1 [35]. The k-dependent dipole moment we use is as follows [12].

Table 1. Parameters describing the conduction and valence bands for $\rm{\alpha -quartz}$ crystal. All values are in atomic units.

View Table

Recent studies have shown the importance of the transition dipole phase and Berry connections on solid-state HHG [22,36]. However, it has been found that when the Berry connections and dipole phase are neglected simultaneously, the parallelly polarized harmonics are reproduced quite well compared to the simulation that includes both effects [37]. Since we focus on the tunneling dynamics along the laser polarization direction, both the Berry connection and the dipole phase are ignored in our simulations.

We consider a linearly-polarized laser field with the vector potential

3. Results and discussion

We first consider the case of a laser field polarized linearly along the GM direction. Figure 2(a) shows the relationship between the initial crystal quasimomentum $k_0$ and the instantaneous momentum at the moment of tunneling ionization ${\bf {k}}({t}^{{r}}_{{s}})$ for different laser intensities with amplitudes $E_0$ ranging from $0.005 \, \rm{a.u.}$ to $0.08 \, \rm{a.u.}$, which corresponds to Keldysh parameters $\gamma =5.24, \, 2.62, \, 1.31, \, 0.66, \, 0.33$, respectively. The results indicate that tunneling ionization does not always occur at the top of the valence band (i.e., at the $\Gamma$ point), which is different from previous assumptions [25,38]. For large $\gamma$, i.e., for nonadiabatic tunneling, ionization at an instantaneous momentum away from the $\Gamma$ point is possible, where the initial velocity of the electron when it appears in the conduction band is nonzero. With decreasing $\gamma$ the distribution of the instantaneous tunneling momenta ${\bf {k}}({t}^{{r}}_{{s}})$ shrinks and gradually approaches 0. For a laser amplitude of $E_0 = 0.08 \, \rm{a.u.}$ ($\gamma =0.33$, see cyan star point line shown in Fig. 2(a)), for all initial $\bf {k}_{\rm{0}}$ the ionization occurs mainly at the $\Gamma$ point, where the initial velocity is approximately equal to 0, which means that the ionization is in the adiabatic tunneling regime. Figure 1(b) schematically shows the pre-acceleration process due to intraband motion before the tunneling process starts in the adiabatic tunneling case. For increased Keldysh parameters, the pre-acceleration process is gradually shortened. For large initial momentum $\bf {k}_{\rm{0}}$, tunneling occurs before the electron is pre-accelerated to the $\Gamma$ point, see Fig. 2(c). Therefore, the initial velocity is nonzero, showing strong nonadiabaticity. For $\gamma =5.24$, see blue circles line in Fig. 2(a), the initial momentum $\bf {k}_{\rm{0}}$ is almost equal to the momentum at the time of tunneling ${\bf {k}}({t}^{{r}}_{{s}})$.

 

Fig. 2. (a) The initial momentum $\bf {k}_{\rm{0}}$ as a function of instantaneous tunneling momentum ${\bf {k}}({t}^{{r}}_{{s}})$ for different laser intensities. (b) The corresponding relationship between the ionization rate and the ${\bf {k}}({t}^{{r}}_{{s}})$. (c) The corresponding ionization rate as a function of the ionization time $t^r_s$ The gray dotted line shows the normalized laser field. (a), (b), and (c) are for the same laser wavelength $\lambda =1000 \, \rm{nm}$. (d), (e), and (f) are for the same laser intensity $I=0.51 \times 10^{10} \, \rm{V/m}$.

Download Full Size | PDF

Consistent with that in atomic gases, the quantum "ionization window" is confined within a fraction of a half laser cycle around the instantaneous maximum of the laser field. This is quite in contrast to the classical picture where ionization happens at any time of the laser field. Figure 2(c) shows the corresponding ionization rate as a function of the ionization time (the real part of saddle point time). The smaller the Keldysh parameter is, the narrower the ionization time window becomes. Since the Keldysh parameter depends not only on the intensity, but also on the wavelength of the laser field, we also compare the results for different laser wavelengths while keep the laser amplitude constant at $0.01 \, \rm{a.u.}$. Similar results can be found by fixing the electric field strength and changing the wavelength of the electric field, see Figs. 2(d)-(f).

The tunneling distance $x_0$ is another important quantity which characterizes the tunneling dynamics and which is valuable for properly describing the electron motion during and after tunneling [8]. Similarly to atoms and molecules, the semi-classical description of the tunneling distance in solids can be obtained evolution along the imaginary part of the ionization time [39]

In the purely adiabatic tunneling regime, it has been predicted that the ionization position can be approximated as a quasi-static ionization position that can be expressed by $x_0=E_{ {g}}/E(t_s)a_0$, with the band gap $E_{ {g}}$ [40]. We show a comparison of the tunneling positions calculated with the saddle point method and the quasi-static approximation in Fig. 3. It can be seen that both tunneling distances decrease as $\gamma$ becomes smaller. For $\gamma =1.3112$ the tunneling exit ranges between $4-5$ lattice cells. This is contrast to the classical trajectory picture of the three-step model for HHG where the tunneling process is ignored, i.e. the initial electron-hole distance after ionization is assumed to vanish [12,20]. Moreover, the tunneling positions calculated with the complex saddle-point method is smaller than that calculated with the quasi-static model, demonstrating that the tunneling process is indeed nonadiabatic. Whereas for $\gamma =0.6556$, the two models provide basically identical tunneling distances implying the tunneling is approaching to the adiabatic regime, as shown in Fig. 3(b).

 

Fig. 3. Comparison of quasi-static ionization positions (blue crosses) with ionization positions by our saddle point method (red circles) for different Keldysh parameters. (a) for $\gamma = 1.3112$, (b) for $\gamma = 0.6556$.

Download Full Size | PDF

4. Conclusions

We investigate the tunneling dynamics for different Keldysh parameters in $\rm{\alpha -quartz}$. We find that for adiabatic tunneling ($\gamma <<1$), the pre-acceleration process of electrons is relevant and electrons with different initial momenta $\bf {k}_{\rm{0}}$ are accelerated to the $\Gamma$ point where tunneling occurs by intraband acceleration. In this case the initial velocity of the electron when it appears in the conduction is approximately zero. In contrast, for nonadiabatic tunneling ($\gamma \sim 1$), the pre-acceleration process is shortened and tunneling at different quasimomenta is possible. Consistent with the situation in atomic systems, in this case the initial velocity of electrons after tunneling is nonzero and the tunneling distance is reduced in comparison to the quasi-static tunneling model. Our method provides intuitive and detailed insights into the kinetic processes of electron tunneling in solids with significant implications for the proper description of strong-field nonlinear phenomena in solids.