Higher-harmonic generation in boron-doped silicon from band carriers and bound-dopant photoionization

We investigate ultrafast harmonic generation (HG) in Si:B, driven by intense pump pulses with fields reaching $\sim 100\mathrm{kV}{\mathrm{cm}}^{-1}$ and a carrier frequency of 300 GHz, at 4 K and 300 K, both experimentally and theoretically. We report several findings concerning the nonlinear charge carrier dynamics in intense sub-THz fields: (i) Harmonics of order up to $n=9$ are observed at room temperature, while at low temperature we can resolve harmonics reaching at least $n=11$. The susceptibility per charge carrier at moderate field strength is as high as for charge carriers in graphene, considered to be one of the materials with the strongest sub-THz nonlinear response. (ii) For $T=300$ K, where the charge carriers bound to acceptors are fully thermally ionized into the valence subbands, the susceptibility values decrease with increasing field strength. Simulations incorporating multi-valence-band Monte Carlo and finite-difference-time-domain (FDTD) propagation show that here, the HG process becomes increasingly dominated by energy-dependent scattering rates over the contribution from band nonparabolicity, due to the onset of optical-phonon emission, which ultimately leads to the saturation at high fields. (iii) At $T=4\mathrm{K}$, where the majority of charges are bound to acceptors, we observe a drastic rise of the HG yields for internal pump fields of $\sim 30\mathrm{kV}{\mathrm{cm}}^{-1}$, as one reaches the threshold for tunnel ionization. We disentangle the HG nonlinear response into contributions associated with the initial photoionization and subsequent motion in the bands, and show that intracycle scattering seriously degrades any contribution to HG emission from coherent recollision of the holes with their parent ions.

Figure 1

Overview of strong-field harmonic generation in Si:B at (a) $T=300$ K—band motion with thermally ionized holes including both band nonparabolicity and energy-dependent scattering; (b) $T=4\mathrm{K}$, whereby light holes are first generated by tunnel ionization of acceptors, followed by significant $\mathrm{lh}\to \mathrm{hh}$ scattering (amplitude of hole trajectories not to scale). (c) Examples of experimental emission intensity spectra for optimal available conditions at $T=300\mathrm{K}$ and $T=4\mathrm{K}$ (measured at the maximum achievable fields with samples A and B, respectively—see text). Relative spectral intensities scaled to aid visual comparison. Note that absolute/optimal HG yields vs $T$ depend on a complex interplay between pump-field strength, dopant density, sample thickness, and the resulting Drude absorption and standing-wave effects for the pump pulse, as addressed in Secs. 3b and 3c. Linear transmitted fundamental spectrum with undoped sample included for comparison.

Figure 2

(a) Experimental transmitted temporal field for a Si:B sample (Sample A: $N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, T=300\mathrm{K}$) for an incident pump peak field of $E_0=115\mathrm{kV}{\mathrm{cm}}^{-1}$ (blue curve), as well as the field of the overtones without fundamental (red) and scaled reference data for an undoped sample (black). Inset shows detail around the pulse peak. (b) Corresponding intensity spectra, as well as spectrum from MC-FDTD simulations with $E_0=100\mathrm{kV}{\mathrm{cm}}^{-1}$ (see text for discussion of pump field scaling between theory and experiment). (c) Experimental and (d) MC-FDTD spectrograms (common color scale as shown). Included in (b) and (c) are dashed curves for the relative intensity and duration, respectively, for ideal nonlinear HG (for the latter, assuming a Gaussian pulse profile, such that $T_n\sim T_1/\sqrt{n}$, curves shown for $T_n=\pm 2T_{\mathrm{FWHM}}$).

Figure 3

(a) Pump field dependence of transmitted harmonic peak fields (Sample A: $N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, T=300\mathrm{K}, L=900\mu \mathrm{m}$). Experimental results (open circles), and MC-FDTD results (filled squares) vs peak pump field ($E_0$, incident external to the sample) for each odd harmonic $n=1–9$. Respective power-law fit to low-field ranges included as straight lines. To aid visual comparison, experimental and MC-FDTD plotted vs different horizontal axis (bottom: experimental data; top: MC-FDTD data with a scaling factor of 0.9), while experimental higher-harmonic fields ($n=5,7,9$) are scaled by the factors given in the labels. (b) Power-law exponents from (a) vs $n$, from experimental and MC-FDTD data, as well as single-point (bulk) MC simulations. Note that the resolvable range of experimental data for $n=7,9$ are already well in the saturation regime. (c) Nonlinear susceptibilities ${\chi }^{\left(n\right)}$ calculated from experimental peak fields in (a) after correction for Drude absorption (see main text)

Figure 4

Local MC results ($N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, T=300\mathrm{K}$). (a) Pump field dependence of relative emission intensities $V_n$ for each odd harmonic $n=1-9$ [solid curves, legend in (d)]. Scaled fluence from MC-FDTD (dotted curves) included for comparison [corresponding to data in Fig. 3]. Black dashed curve corresponds to a linear dependence, for comparison with the curves for $n=1$. Note that local internal pump fields used are $E_0^{\prime }={\widehat{t}}_1{E}_0$ (where ${\widehat{t}}_1=0.46$ is the Fresnel field transmission entering the sample). (b) Time-domain (ensemble-average) velocity for $E_0=110\mathrm{kV}{\mathrm{cm}}^{-1}$ for both hh and lh, as well as scaled pump electric field profile $E\left(t\right)$. (c) Respective fraction of emission intensities from hh/lh for each $n$. (d) Ratio of nonlinear intensity due to $k_b\left(t\right)$ only ($K_{bn}$, see text) relative to emission intensity $V_{bn}$. [(e),(f)] Scattering rates $S\left(t\right)$ during the peak cycle of $E\left(t\right)$ for two peak pump fields ($E_0=55,110\mathrm{kV}{\mathrm{cm}}^{-1}$, respectively) plotted as loci vs the ensemble averages $k_{bx}\left(t\right)$. Also included are the same loci, but vs $k_{bx}·(1+|{\sigma }_{bx}/k_{bx}\left|\right)$ to reflect the extent of the high-energy tail of the distributions. Thresholds for opt. phonon emission for each band (corresponding to an energy of ${\mathcal{E}}_{op}=63.3$ meV) included as vertical lines. Orientation of temporal hysteresis indicated by arrow in (f).

Figure 5

Predicted transmitted (a) fundamental and (b) third-harmonic intensities $I_n\left(L\right)$ from FDTD simulations vs sample thickness $L$ for selected dopant densities $N_d$ ($T=300\mathrm{K}$).

Figure 6

(a) Experimental transmitted temporal field for Si:B sample (Sample B: $N_d=5.0\times {10}^{16}{\mathrm{cm}}^{-3}, T=4\mathrm{K}$) for an incident pump peak field of $E_0=81\mathrm{kV}{\mathrm{cm}}^{-1}$ (blue curve), as well as the field of the overtones without fundamental (red). Inset shows detail around the pulse peak. Note that a temporal window was applied to the wings of the pulse to suppress noise and reflections in the sample. (b) Corresponding intensity spectrum, as well as that from MC-FDTD simulations including tunnel ionization for $E_0=80\mathrm{kV}{\mathrm{cm}}^{-1}$ ($N_d=5.9\times {10}^{16}{\mathrm{cm}}^{-3}$). Both spectra plotted with the same absolute intensity scale. (c) Experimental and (d) MC-FDTD spectrograms. Additional annotations per Fig. 2.

Figure 7

(a) Pump field dependence of transmitted harmonic fluence (Sample B, $N_d=5.0\times {10}^{16}{\mathrm{cm}}^{-3}, T=4\mathrm{K}, L=272\mu \mathrm{m}$: Experimental (open circles, with additional higher sensitivity measurements for $n=3$ as open squares), and MC-FDTD results (filled squares, using $N_d=5.9\times {10}^{16}{\mathrm{cm}}^{-3}$) for each odd harmonic $n=1-9$. Power-law fits for $n=1$ and $n=3$ included as straight dashed lines (exponents $\eta$ as indicated), as well as those for $\eta =3$ and 5 as visual guides for the dependence above the photoionization threshold. See Appendix for comparison of experimental fluence results with those for $T=300\mathrm{K}$ [Fig. 3, for Sample A]. Also included is the predicted fluence for off-resonant HG from bound dopants (green dotted line, using the values for Si:P, see text). [(b),(c)] Spatial profile of peak electric field $E\left(t\right)$ and ionized hole density $N\left(z\right)$ for two values of pump field $E_0=30$ and $80\mathrm{kV}{\mathrm{cm}}^{-1}$, respectively. Only incident field amplitude $E_0$ (without reflected field) shown for $z<0$. Also included are the nominal internal pump fields $E_0^{\prime }=\widehat{t_1}{E}_0$ (used for the local MC simulations when specifying $E_0$).

Figure 8

Local MC results ($N_d=5.9\times {10}^{16}{\mathrm{cm}}^{-3}, T=4\mathrm{K}$). (a) Pump field dependence of relative emission intensities $V_n$ for each odd harmonic $n=1-9$ [solid curves, legend in (c)]. Scaled fluence from MC-FDTD [data from Fig. 7, here dotted curves] included for comparison. As per Fig. 4, the field values $E_0$ given correspond to the external incident fields [see Fig. 7]. (b) Comparison of local MC results for $T=4\mathrm{K}$ in (a) (solid) with those for $T=300\mathrm{K}$ [dashed, see Fig. 4], with common scale. (c) Fraction of emission intensities $r_{\mathrm{hh},n}=V_{\mathrm{hh},n}/(V_{\mathrm{hh},n}+V_{\mathrm{lh},n})$ from hh for each $n$ vs $E_0$ ($r_{\mathrm{lh},n}=1-r_{\mathrm{hh},n}$ omitted for visual clarity). [(d),(e)] Time-dependent hole band populations (relative to $N_d$) for two pump fields $E_0=49\mathrm{kV}{\mathrm{cm}}^{-1}$ and $81\mathrm{kV}{\mathrm{cm}}^{-1}$, respectively. Initial photoionized holes are taken to enter exclusively into the lh band—see text. (f) As per (e), only using a simulation where initial photoionized holes enter exclusively into the hh band for comparison. Vertical scale for (d) at left, for (e) and (f) at right. Insets show magnified ranges.

Figure 9

Local MC results ($N_d=5.9\times {10}^{16}{\mathrm{cm}}^{-3}, T=4\mathrm{K}$). Decomposition of (occupation-weighted) hole acceleration into “intraband” (${\dot{v}}_b^{\left(\mathrm{B}\right)}$) and “interband” (${\dot{v}}_b^{\left(\mathrm{IB}\right)}$) contributions (see text for definitions) for (a) $E_0=49\mathrm{kV}{\mathrm{cm}}^{-1}$ and (b) $E_0=81\mathrm{kV}{\mathrm{cm}}^{-1}$, for each band ($b=\mathrm{hh},\mathrm{lh}$), filtered for each harmonic $n$. Note different vertical scaling in (a) and (b).

Figure 10

Time-dependence of $k$-space distributions from local-MC simulations for (a) $T=300\mathrm{K}$ and (b) $T=4\mathrm{K}$, for both hh (left panels) and lh (right panels), including ensemble average $k_x\left(t\right)$ (parallel to pump field), and rms spreads ${\sigma }_x\left(t\right)$ (along $k_x$) and ${\sigma }_y\left(t\right)={\sigma }_z\left(t\right)$ (along $k_{y,z}$). In (b), data truncated at early times before sufficient photoionized holes exist to perform statistics. (c) Time-dependent distance of hole from parent ion for the case in (b) (hh and lh combined), obtained by integrating $v_g\left(t\right)$ for each particle (608 holes in total, data only for the first 3 ps after respective ionization of each hole, downsampled to a time step of 50 fs for the first 500 fs, and 100 fs thereafter). Color coding for each burst of photoionized holes during each pump half-cycle. (d) Vertical zoom of (c) to allow inspection of small number of holes returning to parent ion during subsequent half-cycles.

Figure 11

Schematic of experimental setup with time-domain electro-optic detection of TELBE THz pulses after transmission through Si:B sample, either at $T=300\mathrm{K}$ (with an additional teflon lens for tighter focusing and incident fields exceeding $100\mathrm{kV}{\mathrm{cm}}^{-1}$) or $T=4\mathrm{K}$ (sample in cryostat). Before sample: Pump field controlled by relative rotation of polarizer-pair; TELBE source-harmonics suppressed by low-pass filter. Following sample: High-pass filter (HPF) used to balance relative intensity of fundamental and harmonics—see Sec. 6b below for details on correction for HPF transmission, and EO response function for the two crystals used.

Figure 12

Measured characteristics of high-pass filter following Si:B sample to avoid saturation of EO signals by fundamental. Transmission (a) intensity and (b) phase, as determined by a combination of THz-TDS and FTIR measurements, including a model fitted curve used for the numerical correction of spectra (based on a complex Pade rational fit).

Figure 13

Calculated relative EO (a) intensity and (b) phase response for the two crystals (see legend) used in the experiments, and used to numerically correct the emission spectra.

Figure 14

(a) Generalized Drude conductivity spectrum for Si:B Sample B used for experiments at $T=4\mathrm{K}$, from MC simulations [at $T=200\mathrm{K}$ using the fitted value of $N_d=5.0\times {10}^{16}{\mathrm{cm}}^{-3}$ in (b)]. (b) Experimental intensity transmission spectrum of sample, and fitted spectrum using the conductivity model in (a).

Figure 15

Experimental intensity spectra and filtered odd harmonics ($n=1-9$) for each pump field $E_0$ ($T=300\mathrm{K}$, Sample A, $N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, L=900\mu \mathrm{m}$).

Figure 16

Normalized experimental time-domain fields for each filtered harmonic in Fig. 15 ($T=300\mathrm{K}$, Sample A, $N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, L=900\mu \mathrm{m}$).

Figure 17

(a) Pump field dependence of transmitted harmonic peak fields (Sample A, $N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, T=300\mathrm{K}, L=900\mu \mathrm{m}$): Experimental results (open circles), and MC-FDTD results (crosses) for each odd harmonic $n=1-9$. Respective power-law fits included as straight lines. (b) Corresponding experimental harmonic fluence $F_n$.

Figure 18

Pump dependence of the relative phases of emitted harmonics, expressed as the temporal shift between intracycle field peaks $\mathrm{\Delta }{t}_n=t_n-t_1$ of each harmonic relative to the fundamental, about the peak of the fundamental envelope. To aid comparison, we also set ${\mathrm{\Delta }}_n\left(E_0\right)\to {\mathrm{\Delta }}_{n-2}\left(E_0\right)$ at each field $E_0$ where successive harmonics $n$ have sufficient signal-to-noise to begin extracting the phase. (a) Experimental, (b) MC-FDTD, and (c) local-MC simulation results. (d) Relative phase delay due to linear propagation ($L=900\mu \mathrm{m})$ assuming a Drude model with constant effective mass vs scattering time $\tau$.

Figure 19

Estimated nonlinear susceptibilities vs pump field $E_0$ for $n=3,5,7$, based on the raw data (dotted lines), correction for Drude absorption (dashed), and fine correction for absorption bleach of the pump at high fields (Sample A, $N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, T=300\mathrm{K}, L=900\mu \mathrm{m}$).

Figure 20

Experimental intensity spectra and filtered odd harmonics ($n=1-9$) for each pump field $E_0$ ($T=4\mathrm{K}$, Sample B, $N_d=5.0\times {10}^{16}{\mathrm{cm}}^{-3}, L=272\mu \mathrm{m}$).

Figure 21

Normalized experimental time-domain fields for each filtered harmonic in Fig. 20 ($T=4\mathrm{K}$, Sample B, $N_d=5.0\times {10}^{16}{\mathrm{cm}}^{-3}, L=272\mu \mathrm{m}$).

Figure 22

Raw experimental spectrum ($T=4\mathrm{K}$, Sample B, $N_d=5.0\times {10}^{16}{\mathrm{cm}}^{-3}, L=272\mu \mathrm{m}, E_0=81\mathrm{kV}{\mathrm{cm}}^{-1}$) using additional high-pass filter to allow inspection of spectral region about the 13th harmonic (dashed rectangle), including $\pm 1\sigma$ confidence intervals.

Figure 23

Experimental emitted harmonic fluence vs peak pump field. (a) $T=300\mathrm{K}$ [Sample A: $N_d=5.75\times {10}^{15}{\mathrm{cm}}^{-3}, L=900\mu \mathrm{m}$, see also Fig. 17 above]. (b) $T=4\mathrm{K}$ [Sample B: $N_d=5.0\times {10}^{16}{\mathrm{cm}}^{-3}, L=272\mu \mathrm{m}$—only data for GaP EO detection, see also Fig. 6 in main paper].

Figure 24

Photoionization rate probability for light holes in Si:B, adopted from [22], used in the MC simulations for $T=4\mathrm{K}$.

Figure 25

Example [(a),(b)] trajectories and [(c),(d)] kinetic energy of photoionized charges vs time for (a) and (c) the ballistic case as assumed in dilute gas phase ($\mathrm{\Gamma }=0$) and (b) and (d) with intracycle scattering ($\mathrm{\Gamma }/\omega =5$) comparable to the solid-state situation for photoionized holes (note the different vertical scales). Each curve begins at its respective ionization time $t_0$. Kinetic energy curves terminated at recollision time.