Ultrafast modification of band structure of wide-band-gap solids by ultrashort pulses of laser-driven electron oscillations

The electric field of high-intensity ultrashort laser pulses substantially perturbs an electron subsystem of a crystal and affects its band structure. Laser-driven oscillations of electrons and holes are frequently referred to as a major mechanism of the perturbation in dielectrics and semiconductors. New physical effects arise when the band structure is modified by an ultrashort pulse of the oscillations driven by a few-cycle laser pulse. Assuming the laser-pulse envelope varies slowly compared to the carrier frequency, we derive analytical relations for the laser-modified band structure by utilizing the Keldysh cycle-averaged nonperturbative approach under the approximation of constant effective mass. Formation of indirect-gap transient bands, suppression of the nonlinear absorption on the leading edge of a laser pulse, and cycle-averaged photocurrent generation driven by the pulse envelope are predicted. Analytical scaling with six laser and material parameters is obtained. The reported results establish the limits of validity of the Keldysh photoionization model and advance understanding of the fundamental effects involved in high-intensity ultrafast laser-solid interactions.

Figure 1

(a) Time variations of electron momentum due to the laser-driven oscillations (lower part) and associated modification of the conduction band (upper part) by the ponderomotive energy of the monochromatic (blue dash-dotted lines) and nonmonochromatic (red dashed lines) oscillations. (b) Expected dynamics of the band $p$ shifts along a momentum direction parallel to the electric field of a laser pulse: from the original direct gap (left lower inset) via two oppositely shifted indirect gap (upper insets), and back to the original direct-gap bands (lower right inset).

Figure 2

Original conduction band (green solid) and transient bands (black, blue, red) evaluated for a Gaussian pulse envelope [Eq. (24); peak intensity 30 $\mathrm{TW}/{\mathrm{cm}}^2$; carrier wavelength 2400 nm; pulse half-width at the level $1/e$ of maximum intensity is 15 fs; zero CEP] at several instants of time for (a) NaCl and (b) AlN crystals under the monochromatic [Eq. (23); dashed lines] and pulse-driven [Eq. (14); dotted lines] approximations. Orange dots on the closed dotted curve and gray dots on the vertical gray dashed line depict positions of the bottom of the transient conduction bands on their laser-driven trajectories for the pulse-driven and monochromatic models correspondingly.

Figure 3

(a) and (b) Normalized envelope of $E$-field pulse (black dotted) and time variations of the total mutual $p$ shift of the energy bands $\delta p_x$ [Eq. (19)] normalized to the half-width of the first Brillouin zone $p_{\text{BZ}}$ for (a) NaCl and (b) AlN. Evaluation for the Gaussian envelope of Eq. (24) is done for two values of CEP: $0^{\circ }$ (red dashed) and ${90}^{\circ }$ (blue solid). The vertical solid lines mark the position of the pulse peak on the time axis for eye convenience. (c) and (d) Band shift $\delta p_x$ normalized to $p_{\text{BZ}}$ vs intensity $I_0$ and CEP ${\varphi }_0$ taken at fixed time instant $t=-{\tau }_p/\sqrt{2}$ for (c) NaCl and (d) AlN. Parameters of the laser pulse are the same for all the panels: peak intensity is 30 $\mathrm{TW}/{\mathrm{cm}}^2$, carrier wavelength is 2400 nm, pulse half-width at the level $1/e$ of maximum intensity is 15 fs.

Figure 4

(a) and (b) Normalized difference between the effective direct band gap and the original band gap for the monochromatic model of Eq. (23) (black curves) and the pulse-driven model of Eq. (21) (red curves) plotted vs time at peak laser intensity 10 $\mathrm{TW}/{\mathrm{cm}}^2$ (dashed curves) and 30 $\mathrm{TW}/{\mathrm{cm}}^2$ (solid curves) for (a) NaCl and (b) AlN. Laser carrier wavelength is 2400 nm. The vertical solid lines mark the position of the pulse peak on the time axis for eye convenience. (c) and (d) Normalized difference between the laser-induced effective and original band gaps vs carrier wavelength for pulse-driven direct [Eq. (21); red solid], pulse-driven indirect [Eq. (20); blue dotted], and monochromatic [Eq. (23); black dashed] effective band gaps plotted at peak laser intensity 30 $\mathrm{TW}/{\mathrm{cm}}^2$ in (c) NaCl and (d) AlN. The effective band-gap values are taken at time instant $t=-15$ fs. Other laser-pulse parameters are the same for all the panels, zero CEP, pulse half-width at the level $1/e$ of maximum intensity is 15 fs.

Figure 5

(a) Normalized difference between the effective direct band gap and the original band gap of AlN evaluated with the different terms of Eq. (21): black dotted—only zero-order terms [i.e., the monochromatic model of Eq. (23)]; green dash-dotted curve 1—zero- and first-order terms; blue dashed curve 2—zero-, first-, and second-order terms; red solid curve 3—the terms of all orders from zero to the third inclusive. Laser-pulse parameters are the same as in Fig. 4, peak intensity is 30 $\mathrm{TW}/{\mathrm{cm}}^2$. (b) Normalized mutual band $p$ shifts along the momentum direction collinear with the laser-pulse electric field evaluated with different terms of Eq. (19): green dash-dotted curve 1—first-order terms; blue dashed curve 2—first- and second-order terms; red solid curve 3—all the terms from zero to the third order inclusive. Laser parameters are the same as in Fig. 3. The vertical lines depict instant of the pulse peak.

Figure 6

(a) A sketch of the instant variations of the electric field (blue solid line) and electron momentum (red dotted line) in the time domain for the (a) zero and (b) $\pi /2$ CEP. Black dashed line depicts the pulse envelope. The time variations of the electron momentum and electric field are normalized so as to follow the same slow envelope.

Figure 7

Illustration of expected influence of the laser-induced $p$ shift of the energy bands on the rate of the photoionization. The data are the same as in Fig. 4.