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Abstract
We report the results of experimental studies and numerical simulation of the dynamics of the electron-hole pairs formation in silicon under the action of a two-period terahertz pulse with a maximum electric field strength of up to 23 MV/cm. It is shown that an inhomogeneous distribution of the charge carrier concentration over the depth of the silicon sample is formed, which persists for several microseconds. This inhomogeneity is formed due to a sharp increase in the rate of filling the conduction band with free carriers in the subsurface input layer of the silicon wafer, which occurs at a field strength above 15 MV/cm.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Studies of the free charge carriers dynamics in semiconductors in a strong electric field of electromagnetic radiation in the far infrared and terahertz spectral ranges are significant interest for controlling the properties of a material, which forms the physical basis for creating ultrafast electronic and optoelectronic devices. The development of this area became possible due to the creation of sources of coherent THz radiation with high intensity [1–4] and the development of methods for its detection [5,6].
The generation of free carriers in silicon induced by subpicosecond THz pulses differs significantly from the generation of electron-hole pairs by femtosecond laser pulses in the optical range of the radiation spectrum. The photon energy in the THz range is two orders of magnitude lower than in the optical range and, in this regard, the main mechanism for the generation of free carriers is impact ionization rather than interband absorption. Studies of the generation of free carriers due to the impact ionization are not only of fundamental interest, but also of great practical importance for the development of materials for solar cells with high efficiency [7] and sensitive photon detectors [8], for understanding the physics of the destruction of dielectrics by femtosecond laser pulses [9], for generation of higher harmonics [10] and for active control of terahertz waves for applications in the THz range, such as THz modulators [11].
The first theoretical and experimental studies of the impact ionization process induced by THz pulses with a pulse duration of tens of nanoseconds were carried out in n- and p-type InSb crystals at a temperature of 78 K [12,13]. With the emergence of THz radiation sources with an electric field strength of up to 1 MV/cm based on femtosecond lasers, a new stage of studies of the impact ionization process in various materials began. In [14], the mechanism of impact ionization in a GaAs crystal induced by THz pulses with an electric field strength of up to 1 MV/cm was studied using exciton luminescence emission in the near infrared region of the spectrum. It was shown that the observed bright luminescence was associated with the multiplication of carriers on the assumption that carriers, coherently controlled by a strong electric field, can reach kinetic energy that is sufficient to increase their density by about three orders of magnitude on a picosecond time scale.
In [15,16], studies of the impact ionization mechanism in silicon were carried out at an electric field strength of a THz pulse of up to 3.6 MV/cm. To achieve such a field, an input pulse with a field of 0.5 MV/cm was amplified by a 200 nm thick gold antenna deposited onto the surface of an experimental sample. It was shown that the ionization rate strongly depends on the initial free carriers density. In particular, the ionization rate increases with a decrease in the initial carrier density. The influence of impact ionization on optical nonlinear effects induced by THz pulses in silicon was observed experimentally in [17,18] The THz pump-probe technique was used in [19] to study the dynamics of free carrier generation in InSb (the band gap is 170 meV, the initial concentration of free carriers was $10^{14}\div 10^{15}$ cm$^{-3}$) due to the mechanism of impact ionization induced by THz pulses with an electric field strength of up to 100 kV/cm at a temperature of 80 K. The concentration of free carriers of $\sim$ 10${}^{16}$ cm${}^{-3}$ was achieved, which corresponded to an increase in the initial concentration by 700%. The dynamics of excited charge carriers in the silicon substrate of THz metamaterial antennas was studied at different wavelengths in the picosecond timescale [20]. Terahertz probe spectroscopy with THz pumping and temporal resolution was carried out using the radiation of a tunable free electron laser in the frequency range 9.3—16.7 THz.
This paper presents the results of experimental studies and numerical simulation of the dynamics of the formation of electron-hole pairs in p-type silicon under the action of a THz pulse with an unprecedently high electric field strength of up to 23 MV/cm. The measurements were carried out in the THz pump – infrared (1240 nm) probe (IR probe) scheme in the geometry of colliding beams.
2. Experimental setup and results
The experimental measurement scheme is shown in Fig. 1.
Fig. 1. Experimental measurement scheme. Si is the silicon sample; DSTMS is the THz generator; PM1, PM2, PM3 are off-axis parabolic mirrors; PD is photodiode.
THz radiation pulses were generated by optical rectification of femtosecond laser pulses in a nonlinear organic crystal DSTMS (4-N, N-dimethylamino-4’-N’-methyl-stilbazolium 2,4,6-trimethylbenzenesulfonate) [21,22]. The DSTMS crystal was pumped by pulses of a femtosecond chromium-forsterite laser system with a radiation wavelength of 1240 nm (IR pump), a pulse duration of 80 fs, and a pulse energy of up to 80 mJ [23] . Radiation from a chromium-forsterite laser system is an optimal pump source for generating THz radiation in a DSTMS crystal with an efficiency of up to 3% and a Gaussian spatial distribution [1,24], which allows focusing a THz beam to a size close to the diffraction limit [25–27]. In the experiments, we used a DSTMS crystal of 8 mm in diameter and 440$\pm$5 $\mu$m thick. The conversion efficiency was 1.6%, and the pulse energy was up to 120 $\mu$J. To absorb the laser pumping radiation, a cut-off filter (LPF8.8-47, Tydex) was installed after the DSTMS crystal, which attenuated the radiation with a wavelength shorter than 34 $\mu$m by a factor of 10$^{8}$. To focus the THz radiation, an off-axis parabolic mirror (PM3) was used with an effective focal length of 50.8 mm and a diameter of 50.8 mm. To achieve the maximum electric field strength, the THz beam was expanded to $\sim$ 50 mm using a telescope consisting of two off-axis parabolic mirrors (PM1 and PM2) with effective focal lengths of 25.4 mm and 152.4 mm. Due to the complete filling of the aperture of the focusing parabola, the THz beam diameter in the focal plane was 240$\pm$10 $\mu$m at the $e^{-2}$ level, which is close to the diffraction limit of 216 $\mu$m (for a central wavelength of 170 $\mu$m). The electric field strength was controlled by changing the pump laser pulse energy using a polarization attenuator consisting of a polarizer and a $\lambda$/2 plate.
For experiments, we used a 235 $\mu$m thick double-side polished wafer of crystalline silicon (100) doped with boron atoms (p-type). The dopant concentration was 1.6$\cdot$10$^{15}$ cm$^{-3}$ according to the resistivity (12.2 Ohm$\cdot$cm) measured by the four-probe method. The sample was mounted on a motorized translator for its optimal positioning in the focal plane of the focusing parabolic mirror. Measurements of the temporal shape of the incident and transmitted by the silicon wafer THz pulses were carried out by the electro-optical detection method using a 200 $\mu$m thick gallium phosphide crystal (GaP (110)).
For the IR transmission measurements, a 100 fs probe pulse with a 1240 nm wavelength propagated towards the THz pump pulse. The use of the counter propagation geometry made it possible to measure the transmission of a silicon layer, the depth of which is determined by the propagation time of a THz pulse in the sample.
The radiation of the IR probe pulse was focused onto a spot of 28 $\mu$m in diameter (at the $e^{-2}$ level) in the central area of the THz spot by a lens with a focal length of 100 mm. The radiation intensity of the probe pulse transmitted through the sample was recorded with an InGaAs photodiode (PD, DET01CFC/M, Thorlabs) with a maximum sensitivity in the range 800 - 1700 nm and a time constant of $\sim$ 1 ns. The probe radiation intensity was 10${}^{11}$ W/cm${}^{2}$, significantly lower than the threshold for the creation of free carriers due to two-photon absorption (the photon energy of the probe pulse is 0.98 eV that is less than the silicon band gap).
The experimental dependences of the transmittance of a silicon wafer at a radiation wavelength of 1240 nm on the propagation time of a THz radiation pulse with different electric field strengths inside the sample are presented in Fig. 2.
Fig. 2. Transmittance of the probing radiation with a wavelength of 1240 nm for a p-type silicon sample with a thickness of 235 $\mu$m as a function of propagation time of THz pulse inside the sample at various values of the maximum electric field strength. Points and lines correspond to experiment and calculations, respectively.
The results of measurements of the temporal shapes of a THz pulse incident with an electric field strength 22.4 MV/cm on a silicon sample and transmitted through it are presented in Fig. 3.
The analysis of the obtained dependences shown in Fig. 2 and Fig. 3 is presented in the next sections.
3. Numerical simulation
3.1 Numerical simulation for carrier concentration
Numerical simulations have been carried out to analyze the processes occurring in silicon under the action of a THz pulse. The dynamics of filling the conduction band with free carriers was calculated using a model that takes into account an increase in the concentration of free carriers upon impact ionization.
To calculate the time dependence of the conduction electrons concentration, we used differential equations [19] in the frame of the Keldysh impact ionization model [28], assuming a quadratic dependence of the impact ionization rate when the energy threshold is exceeded $\varepsilon _{th}\approx \varepsilon _{g}$. Thus, the concentration of free carriers $N _{e,h}$ can be obtained from the system of equations Eq. (1) and Eq. (2):
The computational modeling was carried out in the space-time domain of the variables $t$ and $z$ (the $z$ axis is directed along the normal to the silicon wafer surface). For the numerical solution of the differential equations used in the model, the finite-difference method (FDTD) was applied in the space-time domain on a grid with a discretization step in time $\Delta {t}$ = 0.05 ps, and a step in the spatial coordinate $\Delta {z}=\frac {c}{n_{th}}\Delta {t}$=4.5 $\mu$m. Thus the wafer was divided into several plane-parallel plates (slabs), with a thickness of 4.5 $\mu$m each. The field strength of a THz pulse passing through each slab is assumed to be constant. For calculations, we take the field in each slab to be constant. To fulfill the Bouguer’s law, the intensity decrease occurs at the slabs’ boundaries. Thus, in our model, the THz intensity decreases stepwise rather than exponentially, but envelope of intensity taken at the boundaries fulfill the Bouguer’s law precisely with $\alpha$ being the absorption coefficient. The latter is determined mainly by the carrier density and the frequency of electron-phonon collisions. At each point along the z axis, the maximum value of $\alpha$ is achieved after the pass of the THz pulse.
Than the average integral value should be taken in the calculations $E_{*}(t,z)=E(t,z)(1-e^{-0.5\alpha \Delta {z}})(0.5\alpha \Delta {z})^{-1}$, where $E(t,z)$ is the electric field strength at the beginning of the slab.
As a result of calculations, the dependences of the change in the concentration of free carriers on time in the input slab and the temporal profile of the electric field strength of a THz pulse with a maximum amplitude $E_{max}$=22.4 MV/cm at the entrance of the sample are presented in Fig. 4
Fig. 4. For the maximal field of 22 MV/cm, calculation of the change in the concentration of free carriers (red line for electrons) in the subsurface slab of the silicon wafer and the temporal shape of the THz pulse (blue line for incident on the sample, green line for the transmitted through the sample). The dashed line shows the average time dependence of free carriers concentration.
Equations Eq. (1) and Eq. (2) are symmetrical regarding to electrons and holes. However, initial conditions for them are not symmetrical. For semiconductors, the fundamental ratio holds: $N_e\cdot N_h={n_i}^2$ [31]. In our sample, initial electron and hole density are $N_e=10^5$ cm$^{-3}$ and $N_h=10^{15}$ cm$^{-3}$, respectrively. This means, that when the THz pulse hits the sample, hole-initiated impact ionization strongly predominates. However, each act of the hole-silicon lattice collision results in a $e-h$ pair generation. This leads to a very fast, within 200-300 fs, increase of electron density up to $N_e=10^{11}$ cm$^{-3}$. In further numerical calculations of the dynamics of filling the conduction band, the concentration of $10^{11}$ cm$^{-3}$ was considered to be the initial one. At about 500 fs, electron and hole densities equalize.
The concentration increase rate for the impact ionization process is characterized by the carrier multiplication factor (CM factor), which is defined as the ratio of the maximum concentration of free carriers to the initial one: $K_{CM}=\frac {N(t_1+\Delta {t})}{N(t_1)}$. Then the free carrier‘s generation rate due to impact ionization (ImI) can be determined as $K_{ImI}=\frac {\ln (K_{CM})}{\Delta {t}}=(\Delta {t})^{-1}$ln$(\frac {N(t_1+\Delta {t})}{N(t_1)})$ . For time interval $\Delta {t}$=(1.3$\pm$0.1) ps in the surface slab of the wafer, as a result of the calculation, the values of the carrier multiplication factor and the free carriers generation rate were obtained at various values of the maximum electric field strength of the THz pulse. The data are presented in Table 1.
Table 1. The impact ionization parameters
The distribution of the concentration of free carriers at various values of the maximum electric field strength of the THz pulse along the depth of the slab along the direction of propagation of the THz pulse was calculated slab by slab; the calculation results are presented in Fig. 5.
Fig. 5. Distribution of the concentration of conduction electrons over the depth of the slab after the passage of the THz pulse at various maximum values of the electric field strength.
3.2 Numerical simulation for transmission
In order to simulate a probe pulse transmittance dependences under the THz pulse excitation let us analyze the experimental results (Fig. 2). The minimum carrier concentration, which becomes noticeable in our scheme for measuring the transmission coefficient of a probe pulse, can be estimated from the transmission curves obtained at an electric field strength of 11.4 MV/cm and 13.1 MV/cm. For this estimation we assume that the time dependence of these curves corresponds to the Bouguer law with a constant absorption coefficient. Estimates show that this concentration is $(3\div 5)\cdot 10^{17}$cm$^{-3}$.
Several consequences can be formulated as a result of this estimate.
- (i) IR probe with a wavelength of 1240 nm begins to detect changes in transmission at a concentration of about $10^{17}$cm$^{-3}$. With this, the beginning of the change in transmission at the probe wavelength does not coincide with the beginning of the THz pulse, since it takes some time for the concentration to change.
- (ii) The concentrations of electrons and holes are compared after the concentration of electrons in the excited state reaches the initial concentration of the majority carriers (the concentration of holes in the unexcited state is 1.6$\cdot 10^{15}$cm$^{-3}$) , and this equality remains with a further increase in the concentration with an accuracy to the initial concentration of holes.
- (iii) The distribution of the concentration of excited electrons is highly inhomogeneous over the depth of silicon, and the main contribution to the change in transmission is made by regions of the sample with a relatively higher concentration (Fig. 5).
Based on numerated items, the following model was used.
To calculate the transmission of the probe pulse, we used the Drude model [32] for electrons, which describes the relationship between the absorption coefficient and the electron density and the collision frequency. The frequency $\gamma$ of the electron – phonon interaction in a silicon sample exposed to the THz pulse field was assumed to be equal to $\gamma =2\cdot 10^{14}$Hz [33,34]. This collision frequency $\gamma$ significantly exceeds the frequency $\omega$ of the THz pulse and, therefore, at each instant, the electron velocity is equal to its drift velocity $v_d=\frac {eE_{ins}}{m\gamma }$, where $e$, $m$ are the charge and effective mass of the electron, respectively, and $E_{ins}$ is the instantaneous value of the THz strength vector field. The rate of energy increase $f(E_{ins})$, according to [35], can be represented as $f(E_{ins})=\frac {e^2E_{ins}^2}{2m\gamma }$.
For calculation, the recurrent method was used. The field $E(t,z)$ in the slab, at the current time step, creates an increased concentration of free carriers and changes the absorption coefficient of the material $\alpha$, which will be taken into account in the calculation at the next time step for the next value of the field amplitude entering into the slab. The field $E(t,z)$ entering the slab will be weakened at the output by the current value of the absorption coefficient $\alpha$, that is, the next slab will include the field $E(t,z)e^{-0.5\alpha \Delta {z}}$.
As can be seen from the Fig. 5, the distribution of the concentration of free carriers is inhomogeneous, and the maximum value of the concentration is reached in the input slab of the silicon wafer. Comparison of the temporal shapes of the incident and transmitted THz pulses shows an inhomogeneous attenuation of radiation in the sample (the first half of the pulse is attenuated much less than the second half), which leads to the creation of an inhomogeneous concentration of free carriers along the direction of propagation of THz radiation. The relaxation time of such a concentration distribution over depth is determined by the Auger recombination process and will be, in our case, on average, $10^{-7}\div 10^{-6}$ s [36].
Changes in the distribution of the concentration of free carriers in the sample at a depth of 180-235 $\mu$m are due to the influence of a part of the THz pulse reflected from the inner boundary of the sample. The formation of local concentration maxima is due to the addition of the intensity of the reflected pulse and the intensity of the incident pulse near the exit surface of the sample. with the oncoming movement of the incident and reflected impulses.
The calculated time dependences of the transmission of the probe pulse are obtained. Figure 2 shows them by lines in comparison with experimental data at different values of the maximum electric field strength of the THz pulse at the entrance to the slab.
4. Discussion
Thus, when exposed to a THz pulse, additional free carriers are generated due to the impact ionization mechanism [18,29], which leads to absorption of the probe pulse by free carriers and a decrease in the transmission of the silicon wafer.
The duration of the THz pulse (at the base of the time waveform (Fig. 3)) is 1.5 ps. The leading edge of the THz pulse arrives at the output surface of the silicon wafer in $\sim$ 2.5 ps, but the wafer will remain in the pulse field for another 1.5 ps until the entire pulse is released. As can be seen from Fig. 4, the maximum concentration in each slab will be formed when the last half-period of the THz pulse passes through it, and its largest value (average over the width of the slab) will be in the first input slab.
After the pulse leaves the current slab, the action of the field stops, and the collision frequency begins to decrease at a rate determined by the relaxation parameter of the collision frequency chosen in the model (the relaxation time is about 600-650 fs). At the same time, the local absorption index in the slab begins to decrease and decreases its multiplicative contribution to the overall absorption index of the slab.
The described processes are clearly pronounced for THz pulses with a maximum amplitude higher than 14.0 MV/cm (Fig. 2). For THz pulses with an amplitude that takes smaller values, a sharp drop in transmission with a subsequent rise is not observed. This is due to the fact that the change in the absorption coefficient with depth of the slab in the latter case is much smaller than for the case of THz pulses with large maximum amplitudes. The subsequent drop in transmission in the time interval 3.5–4.0 ps is characterized by an increase in the absorption coefficient caused by the additional action of the THz pulse reflected from the inner surface of the sample. When the THz pulse completely leaves the silicon sample, the absorption index begins to decrease due to the relaxation of the collision frequency [37,38].
Although all experimental data can be explained in terms of free electron absorption of the silicon excited by THz pulses, the fact that impact ionization process is initiated by holes raises a question about the physical nature of the hole-lattice interaction. Since it is not easy to find informal description of this process in literature, we feel it necessary to consider it here.
During hole-initiated impact ionization, a hole in the valence band under the action of a strong external electric field can acquire sufficient kinetic energy ("hot hole") to generate an electron-hole pair. The kinetic energy of a "hot" hole is spent on the excitation of an electron from the valence band to the conduction band (that is, to overcome the bandgap), as a result of which two holes are formed in the valence band, and an electron is formed in the conduction band. Having lost its kinetic energy, the "hot hole" moves on the band diagram to the region of the top of the valence band. In other words, the energy acquired by a set of bound electrons in the valence band (the motion of which can be more clearly described as the motion of a hole) is spent on exciting one of these valence electrons to the conduction band. Thus, this process can be described similarly to electron impact ionization: a hot hole "knocks" a hole out of the conduction band into the valence band, after which that other hole "leaves behind" an electron in the conduction band.
5. Conclusion
The results of experimental studies and numerical simulation of the generation of free charge carriers in the volume of a silicon wafer 235 $\mu$m thick induced by two-period THz pulses with an electric field strength from 10 MV/cm to 22.4 MV/cm are presented. At an electric field strength of up to 14 MV/cm, an almost monotonic change in the transmission of the probe pulse is observed in the time range up to 4 ps. At electric field strengths above 14 MV/cm, a sharp decrease in the transmission occurs within $\sim$ 1.7 ps.
Calculations show that the concentration of free carriers along the direction of propagation of the THz radiation pulse is highly inhomogeneous and, at an electric field strength of 22.4 MV/cm, reaches $\sim$ 10$^{19}$ cm$^{-3}$ in the input surface slab, and $\sim$ 10$^{17}$ cm$^{-3}$ at the exit of the sample. The calculated time dependences of the probe pulse transmittance correlate well with the experimental data.
The possibility of a rapid change in the concentration of free carriers in the bulk of a silicon wafer (depending on the electric field strength of a THz pulse) makes it possible in the future to use the results obtained to create new high-speed optoelectronic devices.
Funding
Ministry of Science and Higher Education of the Russian Federation (075-00892-20-00).
Acknowledgments
The experiments were performed using the unique scientific facility "Terawatt Fem- tosecond Laser Complex" in the "Femtosecond Laser Complex" Center of the Joint In- stitute for High Temperatures of the Russian Academy of Sciences.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
No data were generated or analyzed in the presented research.
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