Ultrafast carrier dynamics in Ge by ultra-broadband mid-infrared probe spectroscopy

In this study, we carried out 800-nm pump and ultra-broadband mid-infrared (MIR) probe spectroscopy with high time-resolution (70 fs) in bulk Ge. By fitting the time-resolved difference reflection spectra [ΔR(ω)/R(ω)] with the Drude model in the 200–5000 cm−1 region, the time-dependent plasma frequency and scattering rate have been obtained. Through the calculation, we can further get the time-dependent photoexcited carrier concentration and carrier mobility. The Auger recombination essentially dominates the fast relaxation of photoexcited carriers within 100 ps followed by slow relaxation due to diffusion. Additionally, a novel oscillation feature is clearly found in time-resolved difference reflection spectra around 2000 cm−1 especially for high pump fluence, which is the Lorentz oscillation lasting for about 20 ps due to the Coulomb force exerted just after the excitation.

(ω: angular frequency, ω p : plasma frequency, Γ: scattering rate, and ε ∞ : permittivity at infinite frequency). Experimental incident angle is set at θ = 45°. For the best fit, the fitting parameter of ε ∞ is between 15 and 18, which closes to the theoretical estimation of ε ∞ = 16 16 . Besides, the time evolution of ω p and Γ can be obtained as in Fig. 3(b). Both ω p and Γ significantly decrease with increasing the delay time and then remain constant for the longer delay time. On the contrary, the carrier mobility μ (= eτ/m*, where is the electron charge, m* is the carrier effective mass, τ is the average scattering time, which is equal to 1/Γ) rises with increasing the delay time due to the reduction of carrier concentration [17][18][19] . After 200 ps, the carrier mobility μ maintains to be ~350 cm 2 V −1 s −1 . Moreover, the photoexcited carriers are only generated near the surface of Ge sample due to the small penetration depth l 800 of 0.2 μ m for 800-nm pump beam (defined as the inverse of absorption coefficient α, where α = 49322.85 cm −1 at 800 nm 20 ). For the ultra-broadband MIR probe beam, the penetration depth is wavelengthand time-dependent. According to l MIR (t) = c/[2n 2 (t)ω], where c is the vacuum light speed, n 2 (t) is the imaginary part of time-dependent refractive index and ω is the MIR angular frequency, the penetration depth l MIR (t) of MIR probe beam is estimated at different delay time. Prior to the pump pulse excitation, Ge is partially transparent (~30%) in the MIR range. The penetration depth of MIR probe beam is only a few μ m at 3 ps after pump pulse excitation. This is much smaller than the sample thickness 500 μ m. However, after 200 ps, it becomes larger than the sample thickness reaching to a few hundred μ m resulting in the appearance of backside reflection feature (R' 2 in Fig. 1) of the sample. Additionally, the detection depth l d of CPU system can be estimated by where T ch = 400 fs is the duration of chirped pulse, θ′ is the refraction angle in Ge (see Fig. 1), n 1 is the real part of refractive index of Ge. Taking n 1 = 4 21 , θ = 45°, and θ′ = 10.2°, the detecting depth l d is 15.2 μ m, which is much smaller than the sample thickness. Therefore, our measurements are free from signal contamination by the backside reflection in the sample. However, l d = 15.2 μ m is longer than the 0.2-μ m penetration depth of pump beam. Therefore, interesting to say that the probe MIR beam can monitor both excited and unexcited regions simultaneously under the present study condition. The detailed analyses of the carrier relaxation processes is discussed in the following sections. Transient carrier diffusion effect. As mentioned in the last section, the photoexcited carriers are generated nearby the surface of Ge (within 0.2 μ m) by pump beam. Besides the short-range collisions among photoexcited carriers, the photoexcited carriers also diffuse from the excited region to the unexcited part due to the spatial gradient of photoexcited carrier concentration. The time evolution of carrier concentration N can be obtained by Eq. (4) 22  By changing the pump fluence (F) from 67 to 135 μ J/cm 2 , the photoexcited carrier concentration increases significantly. However, the photoexcited carrier concentration shows saturation when the pump fluence is further increased from 135 to 202 μ J/cm 2 . To explain the reduction of photoexcited carrier concentration quantitatively, the following differential equation with diffusion term is invoked 24 where D is the diffusion coefficient, G is the carrier generation rate assuming to be much faster than the diffusion rate. By solving Eq. (5), we can obtain the analytic solution as follows, d 0 x Dt 2 4 where N 0 is the total number of photoexcited carriers, which can be determined by integrating N d along 1-dimension depth direction x perpendicular to the sample surface. In the range closed to surface of Ge (i.e. 1 is expected in case the dynamics is only due to the diffusion process. By fitting the data in Fig. 4(b) via Eq. (6) with x~0, the diffusion coefficient can be obtained from the slope, e.g. D = 66 cm 2 /s for the pump fluence of 67 μ J/cm 2 , which is consistent with the theoretical calculation of 65 cm 2 /s 25 . Moreover, the diffusion coefficient D significantly decreases to 20 and 18 cm 2 /s when the pump fluence increases to 135 and 202 μ J/cm 2 , respectively, as listed in Table 1. High-order transient effects. Even though the diffusion model qualitatively reproduces the dynamics of photoexcited carrier concentration especially in the long delay-time range, the difference between experimental data and diffusion model is substantial at shorter delay than 150 ps as shown in Figs 4(a) and 4(b). This implies that other mechanisms might involve in the relaxation processes of photoexcited carriers of Ge in short delay time, such as bandgap renormalization, recombination effect, and intervalley scattering. The bandgap renormalization usually happens after short pulse excitation because of intimate relation between the gap-size and the carrier concentration. However, to observe the bandgap renormalization effect, the measurements of   Fig. 4, Fig. 5 26 , moreover, they propose a clear picture for the roles of reflectance and transmittance, which can provide the information of plasma oscillation and the band absorption, respectively. Therefore, the difference reflection spectra in this study would primarily represent the signals of plasma oscillation rather than the bandgap renormalization, which can be further more definitively neglected in our fittings. Ge is an indirect-bandgap semiconductor material. After photoexcitation, the intervalley scattering from the Γ valley to a side valley dominates the carrier transformation in hundreds of fs 29,30 , and take few μ s for recombination at the Γ point 31,32 . This is the main process for changing the photoexcited carrier concentration in Ge. Especially for the pump in p-type Ge, the relaxation processes from split-off hole band to upper hole band and scattering between heavy hole and light hole bands could be observed 33,34 . However, the carrier relaxation processes inside of the split-off band, heavy-hole band, and light hole band do not induce the changes of photoexcited carrier concentration. Actually, we do observe the reduction of photoexcited carrier concentration in short delay time region, which cannot be simply explained by the diffusion mechanism. Therefore, several other relaxation processes, e.g. the recombination, surface recombination, radiative recombination, and Auger process 35 , should be involved in analysis particularly for high pump fluence as in the following equation, where N is the carrier concentration, D is the diffusion coefficient, γ r is the recombination rate, γ S is the surface recombination coefficient, γ R is the radiative recombination coefficient, γ A is the Auger coefficient, and G is the Gaussian-type generation function for a laser pulse. In order to solve the nonlinear Eq. (7), it is rewritten by the Crank-Nicolson form 36 as described in Supplementary Information. If we simply consider that the photoexcited carriers are just generated or only can be detected on the surface, the photoexcited carrier concentration on surface can be expressed as Additionally, the penetration depth l 800 of 800-nm pump beam is about 0.2 μ m. As mentioned above, the detection depth l d is around 15.2 μ m. This indicates that it is necessary to consider all photoexcited carriers in bulk rather than only on the surface. Therefore, the photoexcited carrier concentration in bulk is expressed as where N(x, t) is the solution of Eq. (7) and α is the absorption coefficient (= 1/l MIR ). The experimental data in Fig. 5 can fit well with Eq. (9) for different pump fluence. When pump fluence increases from 65 to 135 μ J/cm 2 , the substantial decrease in α is clearly shown in Table 1 indicating that a longer penetration depth (l MIR ~ 0.8 μ m) of MIR probe beam for higher pump fluence. Moreover, for the same l MIR , the saturation effect is also found for further increase in the pump fluence to 202 μ J/cm 2 . Based on the well-fit red-dashed lines in Fig. 5, we can further discuss the importance of each term in Eq. (7). For the second term of Eq. (7), the time scale of recombination is in the order of μ s, which is much longer than the measuring range of 400 ps in this study. In the third term of Eq. (7), the surface band bending causes the surface recombination. Without the special surface treatment, the surface recombination velocity is about 1300 cm/s 37 , and its time scale is still in μ s. For the radiative recombination [the fourth term of Eq. (7)], the recombination rate in the bulk Ge with indirect band gap is ~10 −10 cm 3 /s 38 , which is smaller than the commonly found value of the order of 10 −8 cm 3 /s for direct band gap. Thus, the relaxation process of radiative recombination is also negligible in the present experimental condition (the critical value of γ R for this study is 10 −9 cm 3 /s).
As discussed above, the Auger effect dominates the relaxation within 100 ps. The fitting results in Table 1 show that the Auger coefficient γ A (2-3 × 10 −30 cm 6 /s) is independent of pump fluence (F), i.e. the photoexcited carrier concentration (N). According to the relation of 1/τ A = γ A · N 2 38 , we further estimate the recombination time τ A of Auger process, which is in the range of 13-30 ps and dependent on pump fluence. For high pump fluence, e.g. F = 135 and 202 μ J/cm 2 , the τ A becomes small to imply that the efficiency of Auger process would be dramatically enhanced by high photoexcited carrier concentration. On the other hand, the diffusion coefficient decreases down to 20 cm 2 /s with including the Auger process. By the Einstein relation, the value of D/μ at high carrier concentration is ~0.07 39 . Taking μ = 350 cm 2 V −1 s −1 obtained in Fig. 3, thus, the D becomes 24.5 cm 2 /s which is consistent with the fitting results listed in Table 1.

Lorentz force for the photoexcited carriers.
A closer look at the wavenumber dependence of Δ R/R at several delay times in Fig. 6 reveals the fitting of Drude model suffers a significant deviation around 2000 cm −1 , especially for high pump fluence. This implies that some driving forces exist among the photoexcited carriers, which we ascribe to the Lorentz force. Here, we further modified the Drude model with including the Lorentz force, i.e. the so-called Drude-Lorentz model 40 . In Eq. (1), thus, the angular frequency-dependent permittivity is given by where ε ∞ is the permittivity at an infinite frequency, ω is the frequency, ω p is the plasma frequency, Γ is the scattering rate, G s is related to the oscillator strengths, ω 0 is the resonance frequency, and Γ L is the damping coefficient. As shown in Fig. 6, the green-solid lines of Drude-Lorentz model can fit the Δ R/R rather well at different delay time. Interestingly, the difference between Drude model and Drude-Lorentz model, i.e. the Lorentz term, is strongly dependent on the pump fluence and delay time. In the cases of high pump fluence, the Lorentz term becomes more dominate and survives for longer time. From the fitting in Fig. 6, the time-dependent resonance frequency ω 0 can be obtained as shown in Fig. 7. For all pump fluence, ω 0 shows the remarkable red shift below 20 ps.
These results indicate that the photoexcited carriers are bound by a kind of spring force F s = m*ω 0 2 r with distance r. If the Coulomb collision could serve as the spring force, the carrier would be pulled back by the Coulomb force. Even though the paths and directions of collision are random, the motion of carriers can be considered as a simple harmonic oscillation along a specific direction within short delay time. Here, we simply adopted the Coulomb force F C to be the spring force F s , which is just the binding force in Lorentz term. Thus, we have F C = F s + c (where c is a phenomenological proportionality constant), and then the ω 0 can be expressed as where k is the Coulomb's constant, r is the effective distance between the neighboring carriers (which is estimated by n 1/ 3 , and n is the time-dependent carrier concentration), m* is the effective mass, and c is 6 × 10 −11 N. As shown in Fig. 7, Eq. (11) can fit the resonance frequency ω 0 quite well below 20 ps. These results indicate that the oscillating feature of Δ R/R around 2000 cm −1 come from the Lorentz oscillation. Moreover, this Lorentz oscillation is driven by the Coulomb force during the collision among the photoexcited carriers.

Summary
We have studied the photoexcited carrier dynamics in Ge using 800-nm pump and ultra-broadband MIR probe spectroscopy. The time evolutions of carrier mobility, plasma frequency, scattering rate, and carrier concentration have been extracted through the wavelength-(from 200 to 5000 cm −1 ) and time-dependent (below 400 ps) Δ R/R by fitting with the Drude model. For the reduction of photoexcited carrier concentration, the Auger recombination with the Auger coefficient of 2-3 × 10 −30 cm 6 /s dominates the relaxation processes of photoexcited carriers within 100 ps. On the other hand, the long-timescale relaxation process is dominated by the diffusion effect with diffusion coefficient of about 20 cm 2 /s. Moreover, a novel oscillation feature is clearly observed in time-dependent trace of Δ R/R around 2000 cm −1 especially in the cases of high pump fluence, which is considered to be due to the Lorentz oscillation raised by the Coulomb force exerted just after excitation.