Effects of excitation frequency on high-order terahertz sideband generation in semiconductors

We theoretically investigate the effects of the excitation frequency on the plateau of high-order terahertz sideband generation (HSG) in semiconductors driven by intense terahertz (THz) fields. We find that the plateau of the sideband spectrum strongly depends on the detuning between the NIR laser field and the band gap. We use the quantum trajectory theory (three-step model) to understand the HSG. In the three-step model, an electron-hole pair is first excited by a weak laser, then driven by the strong THz field, and finally recombine to emit a photon with energy gain. When the laser is tuned below the band gap (negative detuning), the electron-hole generation is a virtual process that requires quantum tunneling to occur. When the energy gained by the electron-hole pair from the THz field is less than 3.2 times the ponderomotive energy, the electron and the hole can be driven to the same position and recombine without quantum tunneling, so the HSG will have large probability amplitude. This leads to a plateau feature of the HSG spectrum with a high-frequency cutoff at about 3.2 times the ponderomotive energy above the band gap. Such a plateau feature is similar to the case of high-order harmonics generation in atoms where electrons have to overcome the binding energy to escape the atomic core. A particularly interesting excitation condition in HSG is that the laser can be tuned above the band gap (positive detuning), corresponding to the unphysical"negative"binding energy in atoms for high-order harmonic generation. Now the electron-hole pair is generation by real excitation, but the recombination process can be real or virtual depending on the energy gained from the THz field, which determines the plateau feature in HSG.


Introduction
High-order harmonic generation (HHG) results from the interaction of an intense laser with atoms or molecules. HHG provides a mechanism for generating coherent extreme ultraviolet (XUV) and X-ray attosecond pulses [1,2]. The three-step model was established to describe the physical processes of HHG [3,4,5]: The strong laser field tilts the binding potential and the electron escapes from the charged core of the atom or molecule through quantum tunneling; the electron is then accelerated in the free space by the laser field; when the electron recollides with the charged core, a very energetic photon is emitted. Recently high-order teraherz sideband generation (HSG) in semiconductors was predicted [6], which has a physical mechanism similar to HHG but occurs at a very different frequency range. In semiconductors, an electron can be excited from the valance band to the conduction band with a hole left behind. The recollisions between energetic holes and electrons accelerated by a strong THz field result in HSG. Recently, HSG has been experimentally demonstrated [7,8]. The excitonic effect has also been theoretically studied [9].
A fundamental difference between HHG in atoms and HSG in semiconductors is that the electron-hole (e-h) pairs in HSG are elementary excitations caused by NIR lasers. In HHG, the electrons need to overcome the binding energy by quantum tunneling and therefore the quantum trajectories that satisfy the least action condition have only complex solutions. In HSG, the laser frequency Ω can be tuned from below to above the semiconductor bandedge E g , and correspondingly, the initial energy of the e-h pairs generated by the NIR laser can be tuned from negative to positive relative to the bandedge. Particularly, for e-h pairs with positive excess energy (Ω − E g > 0, corresponding to "negative" binding energy of atoms in HHG), the e-h pairs can be directly created by real excitation without quantum tunneling, and in turn the quantum trajectories may have real solutions. On the other hand, the laser can be tuned so high above the band edge that the initial velocities of the e-h pairs are too high for the THz field to drive electrons and holes into recollision. Then recombination of e-h pairs has to occur through quantum tunneling (c.f. quantum tunneling in creation of e-h pairs for laser tuned below band edge). Therefore, there will also be a cutoff of HSG on the excitation laser frequency.
Such consideration motivates us to investigate the dependence of the HSG in semiconductors, in particular, its plateau features, on the frequency of the NIR laser. We calculate the HSG for various excitation laser frequencies and examine the plateau features. The HSG spectrum is explained by quantum trajectories in the three-step model. The previous studies on HSG for negative detuning (Ω − E g < 0) show that the maximal and minimal orders of the sideband plateau are given by the cut-off law [6] where U p = F 2 /4ω 2 0 is the ponderomotive energy, with F and ω 0 being the strength and frequency of the THz field, respectively. The laser detuning E g − Ω plays a role similar to the binding energy in atoms for HHG [6]. The cut-off frequencies can be derived from a classical calculation with the assumption that the electrons tunnel to the conduction band with zero initial velocity. In this paper, we will show that for positive laser detuning (Ω − E g > 0) the HSG plateau cutoffs do not satisfy the Eqs. (1a) and (1b) and depend on the detuning. This characteristic indicates that the HSG spectrum may be modified by tuning the frequency of the excitation laser.

Model and Numerical Simulation
Under an intense THz field, the kinetic energy acquired by the e-h pair can be much greater than the exciton binding energy in the semiconductor and the amplitude of the relative motion can be much greater than the exciton radius, so the essential physics of the HSG can be grasped by the motion of free electrons and holes without Coulomb interaction [7,9]. The equation of relative motion for the electron-hole pair without coulomb interaction is [6] i where p and d are the momentum and the interband dipole matrix element, respectively. A(t) = − F ω 0 sin(ω 0 t)ê z is the vector potential of the intense THz field, and E(t) = E N IR exp(−iΩt) is the NIR laser field. Here we have assumed that the electron and the hole are generated at the same position (r = 0) and the dipole matrix element d is a constant independent of the momentum. The optical polarization of the e-h pair is P(t) = −d * ψ(0, t). The interband polarization can be expressed as where θ(x) is the Heavise step function, and the propagator Using the Fourier transform, we get the polarization strength of the Nth order sideband where the action is Here τ denotes the delay between the recombination and the creation of the e-h pair. The action S(p, t, τ ) is a phase introduced by the motion of the e-h pair with canonical momentum p. Due to the inversion symmetry, only the even-order sidebands (with even N) can be generated [10].
For a strong THz field, the e-h pair performs relative motion with amplitude much greater than its wavepacket diffusion range. Therefore the motion is well grasped by a few quantum trajectories that satisfy the stationary phase conditions. The stationaryphase points are determined by a set of saddle-point equations The physical meanings of these equations are: Eq. The laser detuning Ω − E g determines the mechanism of the e-h generation and recombination (with or without quantum tunneling). If Ω < E g , the electron enters into the continuum via quantum tunneling assisted by the strong THz field. The tunneling physics results in complex solutions of t and τ , which in turn leads to reduced HSG intensity. When Ω ≥ E g , real excitation of e-h pairs is possible and so are real solutions of t and τ . Without requiring quantum tunneling through a binding energy barrier, the HSG can have large amplitudes. However, if the initial energy and hence the initial relative velocity of the e-h pair are too large (Ω − E g > 3.17U p ), the THz field would not be able to drive the electron and the hole back to the same position. Then only through quantum tunneling can the e-h pair recombine, which means the saddle point equations would not have real solutions. So the HSG amplitude will be large when the excitation energy 0 < Ω − E g < 3.17U p and drops rapidly outside this range -the HSG presents a plateau feature not only in the emission spectrum but also in the excitation spectrum. Moreover, as we will show below, the HSG emission plateau depends on the laser excitation frequency.
Substituting Eq. (7a) into Eqs. (7b) and (7c), we get the saddle point equations with respect to t and τ where The equations can have real solutions if Ω > E g and not too much above the band edge, which is not possible for HHG in atoms where the binding energy has to be positive [11]. Equation (8) provides some insight to understand the HSG spectrum. The e-h pair gains or loses kinetic energy to generate sidebands. The dimensionless sideband shift frequency Nω 0 /U p in the plateau regime is shown in Fig. 1 as a function of the delay time for various NIR laser detuning. In our calculation, the effective mass of the e-h pair is chosen to be 0.076 m e as in GaAs, the photon energy of the THz field is 5 meV and its strength is 30 KV/cm. Such parameters above yield U p = 90 meV. As shown in Fig. 1, the upper and lower cutoff orders of the HSG plateau depend on the detuning ∆ ≡ Ω − E g . The saddle point t n and τ n are real if the laser detuning ∆ is such a range that the e-h pair has trajectories satisfying the the classical mechanics. If the laser detuning is outside this range, t n and τ n are complex numbers and the trajectories have to be assisted by quantum tunneling. For positive laser detuning (∆ > 0), the initial relative velocity of the e-h pair is non-zero and can be along different directions. To fulfill the saddle point equation (7a) (the condition that the electron and the hole return to the same position), the initial velocity has to be anti-parallel or parallel to the THz electric field E(t). Therefore for ∆ > 0 the kinetic energy the e-h pair can acquire from the THz field is a double-valued function as shown in Fig. 1. As shown as gaps in Figs. 1 (d)-(e), the real saddle point equations have no real solutions for certain ranges of delay time τ even for laser detuning ∆ ∈ [0, 3.17U p ]. This is because when the initial velocity of the e-h pair is large, the THz field cannot drive the electron and the hole back into the same position unless the delay time is in the proper ranges. For negative detuning (∆ < 0), the e-h pair is generated by quantum tunneling. The saddle-point solutions of τ are in general complex numbers [see Fig. 1 (f,g)]. When the emission frequency outside the plateau region (Nω 0 > 3.17Up or N < 0) goes away from the cut-off frequencies, the imaginary part of the saddle point τ n is large and increases rapidly [see Fig. 1 (g)]. That means small quantum tunneling rate and hence weak sidebands. When ∆ ≥ 3.17U p . When the laser detuning is positive and ∆ ≥ 3.17U p , the initial velocity of the e-h generated by real excitation is so large that the THz field cannot bring the electron and the hole to the same position again for recombination. The saddle-point solutions are also complex. The corresponding saddle point solutions of τ are depicted in Fig. 1 (h) and (i). When the emission frequency goes away from the excitation frequency (blue star symbols) or the band edge (red dot symbols), the imaginary parts of the saddle points of τ [ Fig. 1 (i)] increases rapidly, so the sideband strength P N decreases rapidly and present sharp peaks at the band edge and the excitation frequency (see Fig. 2 (b)).
We now show the quantum trajectory approach to calculating the HSG spectrum. By solving the saddle-point equations above, we obtain the saddle points (t n , τ n ). The action is then where γ = sin(ω 0 τ /2)/(ω 0 τ /2). After integrating the polarization strength P N with Gaussian integral, the susceptibility is determined by [9] Effects of excitation frequency on high-order terahertz sideband generation in semiconductors6 Here S ′′ represents the second-order derivative Jacobian determinant of the corresponding action. The results calculated by exact solution and the saddle-point method are shown in Fig. 2. The characteristic of the HSG spectrum agrees well with the results in Fig. 1: the plateau shrinks when the detuning is increased. The saddle points τ n and t n are complex outside the plateau region, which indicates that the quantum tunneling occurs in the emission. The lower cut-off frequency of the HSG plateau is the band edge of the semiconductor since the final kinetic energy of the e-h pair has to be positive. As discussed above, when the laser detuning is above the 3.17U p threshold, the plateau feature vanishes and instead the sideband intensity presents two rapidly  Fig. 1 dropping peaks at the laser frequency and the band edge. Figure 3 summarizes the width of the HSG sideband as a function of the laser detuning ∆. By searching the real roots of the saddle points τ n and t n for Eqs. (8a) and (8b), we obtain the lower and upper cuttoffs of the HSG spectrum. As shown in Fig. 3, the width of the HSG plateau decreases with increasing of the laser detuning ∆. But the variation is rather slow in the range ∆/U p ∈ (0, 2.2).

Summary
In summary, we have studied how hight-order THz sideband generation in semiconductors driven by intense THz field depends on the detuning of the NIR laser from the semiconductor band edge. As compared with HHG in atoms, the HSG can be studied for positive laser detuning (laser above the semiconductor band edge), corresponding to unphysical "negative" binding energy for HHG in atoms. Exact numerical simulation shows that for positive detuning, the HSG plateau shrinks with increasing the laser detuning and eventually vanishes when the laser is higher than the 3.17U p threshold above the semiconductor band edge. Such features are well understood using the quantum trajectory approach.