Introduction

IMP is an important process induced by Coulomb interactions among electrons in semiconductors1,2. It critically affects carrier transport under high-electric field conditions3 and is of currently studied as a means to improve photovoltaic-device efficiency4,5,6. Extensive studies have been carried out on IMP in semiconductors using a variety of experimental and theoretical techniques7,8,9,10,11,12,13,14,15,16. Traditional experimental studies primarily measure the electric-field dependence of carrier multiplication7. Apart from the difficulty in correlating the electric field with the excess energy Eex of carriers, information concerning state-selective characteristics is completely lost by the energy-integrating nature of this method. Additional spectroscopic studies8,9,10,11,12, which could reduce these drawbacks, neither specify the pertinent energy and momentum states nor determine the absolute rate directly. The rate of IMP has been analytically formulated for a simple case1,17, but there have been no experimental results for comparison with those predicted by this formula, leaving our fundamental understanding of the IMP process ambiguous and incomplete. Therefore, it is critical to measure quantum-mechanical dynamics of IMP on a state-resolved basis.

Here we directly capture state-resolved dynamics of IMP in InSb, using angle- and time-resolved photoemission spectroscopy. InSb crystals with (110) surfaces were excited by fs-laser pulses and photoinjected hot electrons were detected using time-delayed 4.71-eV probe pulses. By detecting photoelectrons as a function of energy ε and emission angle θ, from the surface normal, we directly determine energy- and momentum-resolved distributions of hot electrons at fs-temporal resolution.

Results

Figure 1 (a) shows a photoemission map, at a time delay (Δt) of 50 fs, as a function of ε and θ. The white broken and solid curves are the energy dispersions of the conduction band along the Γ-X and Γ-L lines18. The pump pulse injects electrons at energies of 1.10 eV, 0.76 eV and 0.40 eV above the conduction-band minimum (CBM) from the heavy-hole (HH), light-hole (LH) and split-off (SO) valence band, respectively. Importantly, the transition from the HH band around Λ injects hot electrons predominantly into the L valley. The angle- and energy-integrated intensity of the peaks from the HH band shares more than 80% of total photoemission intensity generated by the pump pulse which creates carriers at a density of 4.2 × 1017 cm−3.

Figure 1
figure 1

Photoemission maps for the nascent electron distributions injected by 1.57-eV excitation and their subsequent dynamics in InSb.

(a) Photoemission image obtained using 1.57 eV excitation pulses and 4.71-eV probe pulses delayed by 50 fs. Excitation and probe pulses are p-polarized. The color scale indicates the photoemission intensity as a function of electron energy and emission angle. The logarithmic intensities are plotted to clearly display regions of weak electron emission. The broken and solid white curves indicate the dispersions of the conduction band along Γ-X and Γ-L directions as a function of emission angle (after Ref. 18). (b) Photoemission image of hot electrons injected into the Γ valley by 1.57-eV light pulses and dynamics of electrons populated near the CBM. (c) Photoemission image and dynamics of hot electrons injected into the L valley by 1.57-eV light pulses. In (b) and (c), the angle-integrated photoemission intensities are plotted as a function of energy and time delay. The color scale indicates the angle-integrated intensity with a linear scale.

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Figures 1 (b) and (c) display angle-integrated intensities of photoemissions from hot electrons and electrons near the CBM as a function of pump-probe delay, Δt. Electrons injected from the SO valence band decay within 200 fs after excitation, while those from LH band decay more rapidly within 100 fs. However, electrons injected into Γ and L valleys from the HH band exhibit relatively slow decay, surviving until ~800 fs. Interestingly, a new peak at Eex = 0.63 eV begins to appear at around Δt = 400 fs. The most striking feature is that the population near the CBM increases dramatically at Δt < 300 fs. As seen in Fig. 1 (b), the electrons injected from the HH band into the Γ valley relax towards low energy states at a rate of 0.2 eV/ps, which is too slow to reach the CBM within 300 fs. Also, low-lying hot electrons from the SO and LH bands do not show any low-energy shift during their respective decays. Therefore, the ultrafast increase in the electron population near the CBM results from fast IMP by hot electrons.

Figure 2(a) displays population dynamics near the CBM. At 293 K, the population increases stepwise within Δt < 300 fs and reaches a maximum at around Δt = 6 ps. The initial finite intensity at Δt < 0 comes from electrons thermally populated at the CBM at 293 K due to the small band-gap energy. The electron density at Δt < 0 as determined using spectral shape analysis is 3.0 × 1016 cm−3 (see supplementary information). Based on this value, we determine that the population maximum corresponds to an electron density of 1.7 × 1017 cm−3 at Δt = 6 ps.

Figure 2
figure 2

Temporal changes in the population near the CBM and two processes of impact ionization.

(a) Temporal changes in the electron population near the CBM measured at 90 K (red line) and 293 K (black line). Angle and energy integrated photoemission intensities (from −5° to 5° and 0.1 to 0.35 eV) are plotted as a function of Δt. The magnitude of the integrated intensity was converted to the electron density based on spectral-shape analysis (see the text). (b) Temporal changes in the intensities of hot electrons in the L valley (blue line) and those populated at Γ(L1) (red line). The same scaling factor to convert the integrated photoemission intensity into the electron density in (a) was assumed for these hot electrons. The thin solid curves are the results of the rate-equation analysis, showing temporal populations of relaxing electrons in the L valley and at Γ(L1). The broken curve is the result obtained by the analysis, predicting temporal change in the population at the L1. (c) Angle integrated photoemission spectra (−5° to 5°) measured at Δt = 1.2 and 2.5 ps. The two peaks at 0.07 and 0.63 eV represent populations near the CBM and at the Γ(L1), respectively. (d) Dispersions along Γ-L direction for valence and conduction bands (after Ref. 18). The black arrow shows the optical transition from LH to the conduction band and pathways of intervalley scattering from Γ-to-L and L-to-Γ are shown by blue arrows. A possible direct impact ionization process is shown by red arrows. The red broken line connecting the L1 and the CBM represents the limit of the direct impact ionization process.

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We can distinguish two distinctive processes in the growth of the population near the CBM: the fast process, at Δt < 300 fs and the slow process which follows it. During the slow process, the hot-electron population decreases in the L valley and a new photoemission band at Eex = 0.63 eV develops. Temporal changes in the intensities of hot electrons in the L valley, angle-integrated for θ > 20° and those populated at Eex = 0.63 eV in the Γ valley are shown by blue and red curves in Fig. 2(b). The first feature represents the energy relaxation towards the L valley (L1) minimum. We note that photoemissions beyond θ = 35° cannot be detected in our detection window, resulting in the observed intensity loss. A recent GW band calculation predicts L1 at 0.61 eV above the CBM18, almost the same energy as Eex where the new band appears. Therefore, we conclude that the new band's appearance is due to the electrons scattered from L1 to the Γ valley at the position, Γ(L1). In fact, the rate equation model successfully correlates the relaxation in the L valley, the accumulation at L1 and the L-to-Γ intervalley scattering and satisfactorily describes the population dynamics at Γ(L1) (see solid curves in Fig. 2(b)). The broken curve shows the predicted changes in the population at the minimum of the L valley.

Importantly, the L-to-Γ intervalley scattering and the increase in population near the CBM takes place in parallel as shown in Fig. 2(c); both peaks near the CBM and electrons at Γ(L1) are increasing concurrently. These results reveal that the hot electrons at the L1 induce both IMP and intervalley scattering competitively. In this IMP process, however, the momentum conservation among electronic states can no longer be satisfied, as no valence states exist on the line connecting L1 and the CBM, as shown by red broken line in Fig. 2 (d). Therefore, IMP by hot electrons at L1 is phonon-assisted2. The red curve in Fig. 2 (a) shows electron population dynamics near the CBM measured at 90 K under the same excitation condition as that measured at 293 K. The enhancement in the slow process is dramatically suppressed at 90 K relative to that at 293 K, supporting our conclusion that the IMP by hot electrons at the L1 is phonon-assisted.

In contrast to the slow process, the fast process is temperature insensitive. A stepwise increase in the population near the CBM is clearly detected at 90 K with almost the same enhancement as observed at 293 K. Considering this result along with the ultrafast temporal response, we conclude that the fast process is due to the direct IMP by hot electrons in the Γ valley. In fact, electrons injected from the SO and LH bands decay within this temporal domain as may be discerned from Fig. 1 (b), with decay times of τd = 150 and 50 fs, respectively. In the direct IMP of hot electrons in the Γ valley, interaction with the LH valence band is crucial1,19,20. As seen in Fig. 2(d), hot electrons in the Γ valley are under ideal conditions to satisfy required energy and momentum conservations because of the approximately linear dispersions of the conduction band and the LH valence band.

Since the electrons from LH band are injected above the energy of L1, they can be scattered into the L valley. To determine the effects of this intervalley scattering on τd, we excited InSb with light pulses at photon energies ranging from 1.2 to 1.57 eV and determined the τd's of wave packets formed by photoinjection. Typical results are shown in Fig. 3(a). The electrons with Eex = 0.53 eV show τd = 150 fs, while those with Eex > 0.65 eV show shorter τd; the decay time of electrons with Eex = 0.79 eV is as short as 50 fs. The relation between the transition rate, given by the inverse of τd and Eex is plotted in Fig. 3 (b). It is clear that there is a critical energy of Eex = 0.63 eV, exactly coincides with the energy of L1, above which τd becomes dramatically shorter. Therefore, hot electrons with Eex > 0.65 eV undergo efficient scattering to the L valley, while those with Eex < 0.60 eV do not.

Figure 3
figure 3

Decay times of hot electrons and relation to excess energy.

(a) Temporal changes in the photoemission intensities from hot electrons injected at Eex = 0.79, 0.65 and 0.54 eV (violet, green and red lines, respectively) plotted on a semi-logarithmic scale. The blue curve, labelled CC, shows the cross correlation trace between pump and probe pulses, which defines the zero time delay. Measurements were made at 293 K. The overlap between pump and probe pulses can be neglected at Δt > 150 fs. Decay times can be determined almost uniquely from the plot, although the analysis for a whole temporal domain including rise time needs more sophisticated methods using the optical Bloch equation (see supplementary information). The thin black solid lines show results of analysis using the optical Bloch equation. (b) The relationship between transition rate, defined by the inverse of the decay time, of hot electrons and excess energy. Green and red dots are for the hot electrons injected from light-hole and split-off valence bands, respectively. The solid curve shows the dispersion of the conduction band along the Γ-L direction (after Ref. 18). The theoretical position of L1 is 0.61 eV above the CBM, while experimentally determined value is 0.63 eV above the CBM as shown by the broken line.

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Discussion

The results described above show that the hot electrons with Eex ≈ 0.5 eV decay with τd = 150 fs. Radiative and phonon-assisted non-radiative recombination occurs at much slower rates21. The possible transient Auger recombination of the hot electrons, induced by photoinjected holes in the LH band, may be ignored as the excitation density of the LH band is at most 1.54 × 1017 cm−3, which is much smaller than the density of states (>1.0 × 1018 cm−3) of the LH band in the energy range where holes are injected. In fact, the decay times of hot electrons with Eex < 0.6 eV did not depend on the excitation density between 2.2 × 1017 and 1.3 × 1018 cm−3. Therefore, we conclude that the decay of these hot electrons is due to direct IMP. Furthermore, as shown in Fig. 3(b), we can determine directly the rate of direct IMP of electrons with Eex = 0.5 eV in InSb to be 1/ τd = 6.7 × 1012 s−1.

The theory of IMP provides the following formula for the rate of IMP for a parabolic system:

where W0 = 4.14 × 1016 s−1, ε0 is the free-space permittivity, ε∞ is the high-frequency permittivity, is the electron effective mass, , is the hole effective mass, Ic and Iv are the cell-periodic overlap integrals, ET is the threshold energy and Eg is the band-gap energy17. In a parabolic system, ET is given by ET = Eg(1 + 2μ)/(1 + μ)1. Using of the LH valence band in InSb, we estimate ET = 0.25 eV at 293 K. When we apply this formula to hot electrons with Eex = 0.5 eV in InSb, we get WIMP = 1.1 × 1012 s−1 even for , establishing an upper bound. Given that Eq. (1) neglects any screening effects1, this theory underestimates the rate significantly. We presume that the discrepancy comes from the parabolic-band approximation in Eq. (1). As seen in Fig. 2(d), the dispersion of the conduction band and LH band is not parabolic, but can be well approximated by a linear dispersion in the energy range from −0.5 to 1.0 eV, except for the gap region. Dispersions along Γ-X and Γ-K directions show the same characteristic18. Therefore, momentum and energy conservations can be satisfied for the four quantum states involved in IMP processes, without any limitations to the initial impacting electronic state except Eex > ET. This would make the rate of IMP significantly larger when compared to the parabolic case, as we observe.

In conclusion, we have demonstrated the state-selective features in IMP processes for the first time and made it possible to compare experimental results and theoretical modeling of IMP rates quantitatively. Our study, based on direct measurements of energy and momentum-resolved electron distributions, provides far deeper insight into the IMP processes in semiconductors than previous studies and gives a sound basis for investigating the current topic of ultrafast interaction of intense THz waves with matter9,10,11.

Methods

Non-doped InSb (100) wafers were cleaved under ultrahigh vacuum conditions (<5 × 10−11 Torr). Surface structures were characterized in situ by a scanning tunneling microscope prior to photoemission measurements22. The (110) surfaces displayed well-ordered (1 × 1)-structure with linear In and Sb rows and a surface-defect concentration less than 0.5%. We used two different femtosecond laser systems. A mode-locked Ti:sapphire laser, operated at a 76 MHz repetition rate, was used to generate fs-laser pulses between 820–730 nm. The fundamental and its third harmonic, generated using beta-barium borate crystals, were used as pump and probe pulses. Temporal widths of the pump and the probe pulses were 80 fs and 100 fs at the sample position in the UHV chamber, giving a cross-correlation trace of 125 fs. To generate shorter pulse widths with wider tunability, we also used a laser system consisting of a Ti-sapphire laser oscillator, a regeneratively amplified Ti:Sapphire laser and a tunable optical parametric amplifier (OPA) operated at 250-kHz repetition rate. The OPA generated 50-fs laser pulses at photon energies tunable from 1.2 to 2.4 eV. A part of the amplified fundamental output at 790 nm was used to generate the third harmonic with a temporal width of 65 fs. Pump and probe pulses, with a preset time delay (Δt), were aligned co-axially and focused on the sample surfaces at 45° to normal.

The ionization energy of InSb(110) is 4.77 eV23. Therefore, the third harmonic of 790-nm fundamental beam (4.71 eV) can probe the states populated in the conduction band, but cannot ionize those in the valence band. This discrimination is very important to capture accurately the photoinjected hot electrons in the conduction band, with completely suppressing intense signals due to direct photoionization from the valence band.

For photoemission spectroscopy, a hemispherical electron analyzer, equipped with an angle-resolved lens mode and a two-dimensional image-type detector served as the electron spectrometer. The entrance slit of the analyzer was placed on the optical plane defined by incoming and reflected light. Two-dimensional images of photoelectrons were recorded as functions of energy and of emission angle θ along the [001] crystallographic direction. Here, θ corresponds to a surface-parallel wavevector k// along the - direction of the surface Brillouin zone (SBZ), on which all states along the Γ-X and Γ-L lines, in the bulk Brillouin zone, are projected with finite values of k//. The instrumental energy resolution with fs-probe light was 45 meV, while angle resolution was limited in the range of ±1°.