Intense-Laser Solid State Physics: Unraveling the Difference between Semiconductors and Dielectrics

Experiments on intense laser driven dielectrics have revealed population transfer to the conduction band to be oscillatory in time. This is in stark contrast to ionization in semiconductors and is currently unexplained. Current ionization theories neglect coupling between the valence and conduction band and therewith, the dynamic Stark shift. Our single-particle analysis identifies this as a potential reason for the different ionization behavior. The dynamic Stark shift increases the band gap with increasing laser intensities, thus suppressing ionization to an extent where virtual population oscillations become dominant. The dynamic Stark shift plays a role dominantly in dielectrics which, due to the larger band gap, can be exposed to significantly higher laser intensities.

Figure 1

Top (a)–(c): Ionization in a semiconductor, with $E_g=0.129\mathrm{a}.\mathrm{u}.$, $\mathrm{\Delta }=0.6\mathrm{a}.\mathrm{u}.$, $a=5.32\mathrm{a}.\mathrm{u}.$, $d=3.46\mathrm{a}.\mathrm{u}.$ for a long (a) and short (b),(c) pulse. Middle (d)–(f): Ionization in a dielectric with $E_g=0.33\mathrm{a}.\mathrm{u}.$, $\mathrm{\Delta }=0.13\mathrm{a}.\mathrm{u}.$, $a=9.45\mathrm{a}.\mathrm{u}.$, $d=5.66\mathrm{a}.\mathrm{u}.$ for a long (d) and short (e),(f) pulse. (a),(b),(d),(e) Final conduction band population versus $\beta =F_{0}a/{\omega }_0$; the shaded region marks where the Keldysh parameter $\gamma =({\omega }_0/F_{0}a)\sqrt{2E_g/\mathrm{\Delta }}<1$. (c) Time evolution for the short pulse in (b) at $\beta =1$. (f) Time evolution for the short pulse in (e) at $\beta =6.3$. The laser field has ${\omega }_0=0.06\mathrm{a}.\mathrm{u}.$ and a Gaussian envelope with width ${\tau }_0=100T_0$ for the long pulse (a),(d) and ${\tau }_0=1.7T_0$ for the short pulse (b),(e). The colors in (a)–(f) show the 2B solution (red), the FVB solution (green), and the GK solution (blue). Further, (d) shows the position of $K_n$ for $n=6$ (purple) and $n=7$ (orange); the colored arrows indicate the $y$ axis to which the line of corresponding color belongs. Bottom (g)–(j): Illustration of the resonance condition in Eq. (5), where ${ϵ}_{\mathrm{ns}}$ (blue lines) is shown for $\beta =0.0$, 1.5, 2.4, 4.2. The solid lines and arrows show the $n=6$ and $n=7$ resonances. The intersection between blue curves and solid lines gives the position of $K_n$; in the absence of an intersection, the channel is closed.

Figure 2

Short pulse time dynamics in the dielectric for the channels centered around $6{\omega }_0$ (dashed), $6.5{\omega }_0$ (dotted), and $7{\omega }_0$ (solid) for (a) $\beta =1.6$ and (b) $\beta =6.3$; for all other parameters, see Fig. 1.

Figure 3

Final dielectric conduction band population for the short pulse versus ${\mathrm{\Omega }}_r/E_g$, with $\beta =6.3$ for the 2B (solid) and FVB solution (dash dot); for all other parameters, see Fig. 1; ${\mathrm{\Omega }}_r$ is changed by varying the dipole moment $d$. The dashed line indicates ${\mathrm{\Omega }}_r/E_g$ used in Fig. 1.