Role of localized electronic states in high-order harmonic generation from doped semiconductors

We investigate high-order harmonic generation (HHG) in sparsely doped semiconductors. The doping in semiconductors breaks the periodic translation invariance in space and introduces localized electronic states (LESs), which leads to two additional electron transition channels, namely, the nonvertical transitions and transitions from LESs. By involving these channels, one can intuitively understand the enhancement of the harmonic yield and the extension of the cutoff energy found in previous works. Moreover, the transition from LESs is space localized and shows self-probed behavior under strong laser fields, encoding the structure information into the HHG. To demonstrate this, we analyze the HHG process in a doped semiconductor with a pair of impurities, where the recombination process of the electron from LESs can be interpreted as a two-center interference. This imprints the internuclear separation between the impurities into the minima of the harmonic spectra. Our work reveals the underlying physical mechanisms in HHG from doped semiconductors and suggests all-optical metrology for structural and dynamic information of LESs.

Figure 1

The band structure of (a) the intrinsic semiconductor and (b) donor-doped semiconductor. The horizontal line in (b) indicates the energy of the LES. The VT, NT, and LEST channels are indicated by the blue, red, and green arrows, respectively. The VT channels between VB and CB1 (CB2) are labeled VT1 (VT2). The VT channels between CB1 and CB2 are labeled VT3. The LEST channels between the LES and CB1 (CB2) are labeled LEST1 (LEST2).

Figure 2

(a) The TDM for VT1, the NT, and LEST1. The blue curve shows the magnitude of ${\mathit{d}}_{v{c}_1}\left(\mathit{k}\right)$ for VT1 in the intrinsic semiconductor. In the donor-doped semiconductor, the red curve shows the magnitude of ${{\mathit{d}}^{\prime }}_{v{c}_1}({\mathit{k}}^{\prime },\mathit{k})$ from the top of the VB (${\mathit{k}}^{\prime }=0$) to CB1 states, the green curve shows the magnitude of ${\mathit{d}}_{c_{1}0}\left(\mathit{k}\right)$ for LEST1. (b) The logarithm of the magnitude of the TDM ${{\mathit{d}}^{\prime }}_{v{c}_1}({\mathit{k}}^{\prime },\mathit{k})$, i.e., ${log}_{10}\left[\right|{{\mathit{d}}^{\prime }}_{v{c}_1}({\mathit{k}}^{\prime },\mathit{k})\left|\right]$, in the donor-doped semiconductor.

Figure 3

The harmonic intensity contributed by the VT (blue curve), NT (red curve), and LEST (green curve). The black dotted curve and the gray shaded area indicate the total harmonic intensity. The arrows indicate the cutoff energies at 4.8 and 18.8 eV.

Figure 4

The harmonic spectra obtained by solving the TDSE. The harmonic spectra for the donor-doped and intrinsic semiconductors are shown. The red and blue arrows indicate the cutoffs of donor-doped and intrinsic harmonic spectra, respectively.

Figure 5

The TDPI pictures for (a) the intrinsic semiconductor and (b) donor-doped semiconductor. The colors indicate the populations of electrons in the VB, LES, CB1, and CB2. The isolated line between the VB and CB1 in (b) corresponds to the LES. $T_c$ is the optical cycle. For clarity, the electron population in CB2 for the intrinsic semiconductor is multiplied by 100.

Figure 6

(a) The potential of the donor-doped semiconductor with a pair of impurities. (b) The logarithm of the electron density distribution in the LES. (c) The band structure of the doped semiconductor. The horizontal dashed line in the band gap indicates the energy of $|{\varphi }_0\rangle$. (d) The TDM for the LEST between $|{\varphi }_0\rangle$ and CB states. The length of the arrows indicates the amplitude of the TDM, and the direction of the arrows indicates the direction of the TDM.

Figure 7

(a)–(d) show the magnitudes of the TDM $|{\mathit{d}}_{c0}\left(\mathit{k}\right)|$ with different internuclear separations: (a) $R=2a_0$, (b) $R=3a_0$, (c) $R=4a_0$, and (d) $R=5a_0$. (e)–(h) show the parts of harmonic spectra corresponding to the TDM. The arrows indicate the minimums of the harmonic spectra and the corresponding minima of $|{\mathit{d}}_{c0}\left(\mathit{k}\right)|$.

Figure 8

(a) The parts of the harmonic spectra corresponding to the LEST between the LES and CB states. The results for $R=1.7a_0,1.9a_0$, and $2.2a_0$ are shown. The arrows indicate the minima of the harmonic spectra. (b) The comparison of the theoretical $R$ and $R$ obtained by the positions of the minima using Eqs. (29) and (30). We choose 13 states with internuclear separations around $R=2a_0$. These states are numbered in ascending order of the internuclear separations. The horizontal coordinate in (b) is the number of states. The black line indicates the theoretical values of $R$. The blue circles indicate the obtained $R$ with laser intensity $I_1=0.36\mathrm{TW}/{\mathrm{cm}}^2$, and the red crosses indicate the results with laser intensity $I_2=1.42\mathrm{TW}/{\mathrm{cm}}^2$. The gray shaded area indicates the maximum deviation from the theoretical value is 2 Å.

Figure 9

(a) The TDM of the acceptor-doped semiconductor for the VT, NT, and LEST. The blue curve shows the magnitude of ${\mathit{d}}_{v{c}_1}\left(\mathit{k}\right)$ for VT1, the red curve shows the magnitude of ${{\mathit{d}}^{\prime }}_{v{c}_1}(0,\mathit{k})$ for the NT, and the green curve shows the magnitude of ${\mathit{d}}_{c_{1}0}\left(\mathit{k}\right)$ for LEST1. (b) The TDPI picture for the acceptor-doped semiconductor. The colors indicate the populations of electrons in the VB, LES, CB1, and CB2.

Figure 10

The harmonic spectra for the doped semiconductor with a pair of impurities. The simulation is performed with all the occupied states in the VB and the LES as the initial state. The intensity and wavelength of the driven laser are shown in the plot. The arrows and dashed lines indicate the minima in the harmonic spectra.