Single-beam lifetime measurements via self-induced optical absorption

Kevin M. W. Boyd; Rafael N. Kleiman

doi:10.1364/OE.27.004445

1. Introduction

The pump/probe technique is an indispensable characterization tool that finds application across a diverse set of physical sciences. Pump/probe studies with ultrafast lasers have been used to probe the fundamental dynamics of chemical and atomic systems from picosecond to femtosecond to even attosecond resolution [1–3]. In the biological sciences, pump/probe techniques have been developed to measure the temperature of cancer cells [4], or perform time-resolved crystallography on biological macromolecules [5]. In semiconductor physics, pump/probe studies have been used to measure the fundamental thermalization dynamics of electrons [6], to examine the built-in electric field at the surface [7], or to analyze impurities via the recombination lifetime [8]. A generic pump/probe experiment involves a pump source which excites some physical phenomenon, and a probe source which monitors the evolution of that phenomenon over time. Typically these sources are separate laser beams, or a single source split into two components. Ultrafast studies that require sub-nanosecond timing resolution use pulsed lasers for the pump and the probe where the pulse width is typically much shorter than the decay time of the phenomenon being studied. The necessary temporal resolution is achieved by delaying the arrival time of the probe beam with respect to the pump. Alternatively, ultrafast timing resolution may be achieved by optical heterodyning of two lasers that can be tuned in wavelength with respect to one another [9]. Pump/probe studies of phenomenon that decay on longer timescales often use a pulsed or modulated pump laser and a CW probe to resolve the decay [8,10]. In this case, the temporal resolution is achieved with wide modulation bandwidth for the pump and fast electronic detection for the probe. Often the pump and probe are widely separated in wavelength to decouple the excitation of the phenomenon from the monitoring of it. For example, complex proteins are excited with an optical pump and their crystallographic structure is studied with an x-ray probe [5]. Separate wavelengths for the pump and probe are necessary since it is not possible to elucidate crystallographic information from an optical probe.

In some cases, the phenomenon being excited will itself act upon the excitation source. For example, when light from a laser is absorbed in a semiconductor to excite free electrons and holes, these free-carriers themselves will induce further absorption of the laser. By exploiting this self-induced absorption it is possible to extract information equivalent to traditional dual-beam modulated pump/probe techniques. The free-carrier density is commonly measured in a dual-beam modulated pump/probe experiment where the pump and probe beams have energy greater than and less than the bandgap of the semiconductor, respectively. This arrangement naturally decouples the band-to-band absorption used to generate free-carriers from the free-carrier absorption (FCA) used to analyze their density. However, since the pump beam also undergoes FCA we show that the use of a separate probe beam is redundant. The absorption in a single beam has a component that varies linearly with power, and a component that varies quadratically. The linear component is due to band-to-band absorption which results in the excitation of free-carriers [11]. The density of free-carriers that is injected is directly proportional to the incident power. The quadratic component arises from free-carrier absorption, where free-carriers attenuate the incident light via intraband absorption of electrons in the conduction band and holes in the valence band. The magnitude of free-carrier absorption is directly proportional to incident power and to the free-carrier density, so the free-carrier absorption in a single beam varies quadratically with incident power. This quadratic component arises due to the self-induced absorption of the laser beam.

If the non-linear component of the pump absorption can be isolated from the linear component, then the experimenter can access the exact same information that would be acquired using a separate probe beam. Mathematically, the transmission of light through a semiconductor wafer is given by exp(–αW – α_FCAW), where α and α_FCA are the band-to-band and FCA coefficients, respectively, and W is the wafer thickness. The implementation of self-induced absorption is optimal when the criterion αW ~1 is met. This simultaneously ensures that a sufficient fraction of the beam is absorbed in the wafer in order to excite free-carriers, while also ensuring a non-negligible fraction transmits through the wafer to be collected and analyzed. It is also required that α_FCA << α so that the presence of injected free-carriers does not strongly perturb the injection due to band-to-band absorption. Since α is a strong function of wavelength in a semiconductor, by adjusting the wavelength we can always satisfy αW ~1. The criterion α_FCA << α is almost always satisfied since FCA is very weak at optical frequencies.

In this work we develop single-beam lifetime measurements via self-induced absorption as an alternative technique to dual-beam modulated pump/probe method. For semiconductor characterization, it is the time (or frequency) resolved behavior of the free-carrier density that is of interest to experimenters since it provides direct access to the electronic properties of the semiconductor, such as the diffusion coefficient, surface recombination velocity, and bulk recombination lifetime [8,12]. We utilize single-beam self-induced absorption to measure the effective recombination lifetime of a silicon wafer, and show that it extracts an equivalent lifetime value as dual-beam modulated pump/probe. There are several reasons, both fundamental and practical, for why silicon is the ideal platform for the development and demonstration of the single-beam self-induced absorption technique. Firstly, silicon has an indirect bandgap so light with energy close to its bandgap is only weakly absorbed and we can readily satisfy the αW ~1 criterion. Incidentally, the fundamental emission of the Nd:YAG laser (λ ~1064 nm), which is a standard and mature laser technology, is just above the band-edge of silicon (λ ~1100 nm), in energy. Secondly, the recombination lifetime of silicon is relatively long (> 1 μs) compared to direct bandgap semiconductors such as GaAs (~1–100 ns). Since the experimental apparatus requires bandwidth comparable to the inverse recombination lifetime in order to resolve the lifetime, a study of silicon is less demanding on the bandwidth of the experimental apparatus than other semiconductors. Finally, there is utility in experimental techniques that measure the lifetime of silicon. The recombination lifetime is a valuable metric for the photovoltaics community since it can be used to quantify the impurity content of semiconductor feedstock used for solar cells [13] and for solar cells at various stages of fabrication.

Free-carrier recombination in a semiconductor occurs via radiative, Auger, and trap-assisted processes [13]. Typically, trap-assisted recombination is the dominant mechanism in silicon at low carrier densities (< 10¹⁸ cm^–3). Trap-assisted recombination is a combination of two mechanisms that occur in the volume and at the surface of the semiconductor. Recombination in the volume occurs through electrically-active impurities in the semiconductor lattice, whereas recombination at the surface occurs through energy states originating from dangling bonds. All of the various recombination processes occur simultaneously, and each is characterized by a recombination lifetime that depends on both intrinsic and extrinsic material properties of the semiconductor. The net rate of recombination is the sum of the individual rates of recombination and is characterized by an effective lifetime which is the reciprocal sum of the individual lifetimes. It is the effective lifetime which is observed directly in an experiment. It is possible to decouple the individual lifetimes that constitute the effective lifetime by examining the temperature and/or injection-level dependence of the effective lifetime [13]. However in this work we seek only to demonstrate that our technique can measure the effective lifetime. Interpretation of this quantity to isolate the lifetimes associated with the various recombination mechanisms is beyond the scope of this work.

The reduction of an experiment to a single laser beam is a dramatic simplification of traditional dual-beam modulated pump/probe studies of semiconductors. In any dual-beam technique, care must be taken to ensure that the pump and probe sources overlap on the sample. This is straightforward to do on a laboratory bench, but if the study is to be performed in situ where the beams have to propagate over a long distance this can become prohibitively difficult. Robust alignment of both sources is required to prevent systematic drift of one beam with respect to the other over the duration of the experiment. When beam diameters are small (for silicon the pump and probe diameters are on the order of millimeters [8]), high-precision optical components are necessary in order to ensure the pump and probe beams remain overlapped. Power, polarization, and beam diameter are all quantities that an experimenter might need to control for a measurement. In dual-beam studies both sources require separate supporting optics. The second beam itself and the additional optical components required to support it increase the size and cost of the experimental apparatus. The single-beam self-induced absorption technique also has advantages over traditional silicon lifetime metrologies. Microwave Photoconductance Decay (μ-PCD) and Quasi-Steady State Photoconductance (QSSPC) are two commonly used techniques for measuring the lifetime of silicon wafers, especially for photovoltaic applications [14,15]. μ-PCD analyzes the decay of free-carriers by measuring the time-resolved change in microwave reflectance due to photoconductivity in the wafer, whereas QSSPC measures the steady-state free carrier density via an RF inductance bridge. Though both techniques are technically non-contact, the wafer must be positioned inside a microwave cavity for μ-PCD or in close proximity to the eddy current sensor for QSSPC. The spatial constraints of $μ$ -PCD and QSSPC reduces the flexibility for implementing them in situ. This problem is partially alleviated with the dual-beam modulated pump/probe method since it is a purely optical technique and further improved and simplified by the single-beam self-induced absorption technique.

As described above, the transmitted power of a laser has a linear and quadratic dependence on the incident power, where the quadratic dependence arises due to self-induced absorption of the laser. Isolation of the weak quadratic term usually requires high power levels such that the quadratic term becomes comparable to the linear one [16,17]. In this work, we develop the theory for the self-induced absorption technique by analyzing the power of a modulated above-bandgap laser passing through a slab of semiconductor material. We show that the transmitted power at the modulation frequency also has a linear and quadratic dependence on the incident power, however the quadratic dependence can be made manifest at much lower power levels though an isolation and normalization procedure that we describe below. The quadratic component is related to the recombination lifetime of the semiconductor, which we seek to measure. We validate our model experimentally using a Nd:YAG laser emitting at 1064 nm as the optical source, and show that the recombination lifetime that is measured is in agreement with the lifetime measured by a dual-beam modulated pump/probe setup that uses a separate 1550 nm laser as the probe. Furthermore we validate that, after signal linearization, the FCA component varies linearly with incident power as expected. The slope of this linear dependence is proportional to the FCA cross-section of the semiconductor, which is a material constant. The constant that we measure is in agreement with values from the literature, providing further validation of the model. We conclude by examining some potential applications for the single-beam self-induced absorption technique.

2. Experimental

In this study, we demonstrate the single-beam self-induced absorption technique by measuring the recombination lifetime of a monocrystalline silicon wafer. The wafer is double-side polished, phosphorus doped (N_d = 4.5 × 10¹⁴ cm^–3), (100) oriented, Czochralski grown, and 1.50 mm thick. The wafer surfaces are passivated by native SiO₂ layers ~2 nm thick. A diagram of the experimental apparatus used to implement the single-beam technique is shown in Fig. 1. The pump is a Laser Quantum Opus CW laser emitting 2W of optical power at 1064 nm. The 1064 nm radiation is weakly absorbed in silicon (αW ~1) and so a fraction of the beam will transmit through the wafer. This is necessary so that the transmitted beam can be collected and analyzed. The beam is expanded with a Standa Beam-Expander to a diameter of 7.7 mm. The laser is modulated by a custom Conoptics Electro-Optic Modulator (EOM) which is driven by a Conoptics Model 25A driver, capable of impressing an analog signal onto the laser beam at frequencies ranging from DC to 25 MHz. The drive signal for the modulator is sinusoidal, and supplied by a Zurich Multi-Frequency Lock-In (MFLI) amplifier where the modulation depth (defined in the section below) is ~63%. After transmitting through the wafer, the laser is focused into a Newport Model 2033 Germanium detector set to 2000 × gain. The detector signal is demodulated by the Zurich MFLI at the modulation frequency ω. Half-waveplate/polarizer pairs before and after the sample control power into the sample and detector, respectively.

Fig. 1 Schematic Diagram of Experimental Apparatus.

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In order to validate the results of single-beam self-induced absorption, the traditional dual-beam modulated pump/probe technique is implemented in situ. We use the same 1064 nm laser as in the single-beam measurement to serve as the pump beam for the dual-beam measurement. The probe beam is a Thorlabs FPL1009S fiber optic laser emitting at 1550 nm and driven by a LDC205C current source set to 100 mA, which results in about 25 mW of optical power. A fiber coupler is used to collimate the laser beam upon exiting the fiber, and then a beam expander is used to expand the probe beam to a diameter of about 10 mm, which fully covers the area illuminated by the pump beam. This is to ensure that the free-carriers generated by the pump do not escape the probe beam due to radial diffusion in the plane of the wafer. The probe beam is then focused into a second Newport Model 2033 detector set to 2000 × gain. The pump and probe laser propagate orthogonally and parallel to the optical table. The sample holder is rotated so that the probe illuminates the sample at Brewster’s angle (approximately 75° from normal at 1550 nm for silicon), leading to a 15° angle-of-incidence for the pump beam. Single and dual beam data are collected in separate frequency sweeps, under nearly identical experimental conditions.

The strategy for collecting single-beam data is decidedly different from that of dual-beam data. In dual-beam modulated pump/probe measurements the signal is entirely due to the 1550 nm probe beam since the 1064 nm pump beam is rejected from the detector. The signal is collected directly and does not require normalization against a background spectrum. In this case, the signal-to-noise ratio of the signal is controlled by the time-constant on the lock-in amplifier, which can be adjusted to achieve an acceptable level of noise suppression. For the dual-beam data collected in this work, the time constant is set to 100 ms. In order to isolate the FCA component of the single-beam signal, the sample spectrum must be normalized against a background spectrum collected without a sample in place. Naturally these spectra are collected in separate measurements since they cannot be measured simultaneously. The normalization procedure is limited by the fidelity with which an individual frequency spectrum can be repeatably measured. The collection of a frequency spectrum is a serial process that involves demodulating the detector signal one frequency at a time until the whole spectrum has been swept out. Serial collection is susceptible to drift error, especially when acquisition times are long. In this case, we cannot achieve the desired signal-to-noise ratio simply by adjusting the time-constant for demodulation, since long time constants lead to long acquisition times. Instead we set the time-constant to 10 ms so that individual frequency spectra can be collected on a time scale that is short with respect to experimental drift. By collecting multiple spectra we can then average out random noise that arises due to the short time-constant. With this strategy we are able to achieve a detection floor that is comparable to the resolution limit of the 14 bit analog-to-digital converter on the lock-in amplifier.

3. Theory

Consider the transmission of a harmonically modulated, monochromatic, Gaussian laser beam through a semiconductor wafer of thickness W. The total transmitted power is given by [18]:

P (t) = \int_{0}^{\infty} T \frac{2 P_{0}}{π w^{2}} e^{- \frac{2 r^{2}}{w^{2}}} (1 + m \cos ω t) e^{- α W - α_{F C A} W} (2 π r d r) .

Here T is a generalized transmission coefficient of order unity that accounts for multiple reflections inside the semiconductor, P₀ is the incident power of the laser, w is the radius of the laser beam defined by the point where the intensity is 1/e² of its peak value, ω is the angular modulation frequency, m is the modulation depth of the laser, α is the band-to-band absorption coefficient at the laser wavelength, α_FCA is the free-carrier absorption (FCA) coefficient at the laser wavelength, W is the wafer thickness and r is the radial component of a cylindrical coordinate system. The exponential with variable r describes the profile of a Gaussian laser beam. The integral sums up the power contributions from each differential area 2πrdr.

Since FCA is small, α_FCAW << 1 and the related exponential can be approximated by a Taylor series expanded to first order:

P (t) = \int_{0}^{\infty} T \frac{4 P_{0}}{w^{2}} e^{- \frac{2 r^{2}}{w^{2}}} (1 + m \cos ω t) e^{- α W} (1 - α_{F C A} W) r d r .

The FCA coefficient is given by α_FCA = σ_FCAn(r,t) where σ_FCA is the free-carrier absorption cross section and n(r,t) is the density of free-carriers generated by optical injection. The free-carrier population consists of a DC and AC component, so we can write n(r,t) = exp(–2r²/w²)[N_DC + N_AC cos(ωt – ϕ_FCA)]. Expressions for N_DC and N_AC are found by solving the continuity equation for free-carriers in the semiconductor for the case when the optical generation rate of free-carriers varies harmonically. We find that N_DC and N_AC are given by (see Appendix):

N_{D C} = \frac{2 P_{0} λ f_{a} τ}{h c π w^{2} W},

N_{A C} = \frac{2 P_{0} λ f_{a} τ}{h c π w^{2} W} \frac{m}{\sqrt{1 + ω^{2} τ^{2}}},

where f_a is the fraction of laser power absorbed by the semiconductor, h is Planck’s constant, c is the speed of light, λ is the wavelength of the laser, and τ is the effective recombination lifetime of the semiconductor, which is the parameter of interest in this work. The fraction f_a of optical power that is absorbed in the wafer is measured by accounting for the total power reflected from and transmitted through the wafer. The free-carrier population takes on the same radial dependence as the excitation laser since the beam radius is assumed to be much larger than the diffusion length of free-carriers. Substituting α_FCA and n into Eq. (2) and integrating over the radial dependence:

P (t) = T P_{0} (1 + m \cos ω t) e^{- α W} (1 - \frac{1}{2} σ_{F C A} (N_{D C} + N_{A C} \cos (ω t - ϕ_{F C A})) W) .

Multiplying out (5) and collecting terms of like time dependence:

\begin{matrix} P (t) = T P_{0} e^{- α W} [1 - \frac{σ_{F C A} W}{2} N_{D C} - \frac{σ_{F C A} W}{4} m N_{A C} \cos ϕ_{F C A} \\ + (1 - \frac{σ_{F C A} W}{2} N_{D C} - \frac{σ_{F C A} W}{2 m} N_{A C} \cos ϕ_{F C A}) m \cos ω t - \frac{σ_{F C A} W}{2} N_{A C} \sin ϕ_{F C A} \sin ω t \\ - \frac{σ_{F C A} W}{2} m N_{A C} \cos ϕ_{F C A} \cos 2 ω t - \frac{σ_{F C A} W}{2} m N_{A C} \sin ϕ_{F C A} \sin 2 ω t] . \end{matrix}

Equation (6) is the power transmitted through a semiconductor when the laser beam undergoes both band-to-band and free-carrier absorption. There is a DC component, as well as AC components at ω and 2ω. The DC and ω terms contain factors unrelated to FCA, whereas the 2ω terms are directly proportional to the FCA cross section. The 2ω term arises from the frequency mixing of the ω component of the laser beam and the ω component of the free-carrier population. In this work we choose to extract the signal at ω instead of 2ω. In principle, the 2ω component is the most natural term to isolate since it is directly proportional to the FCA term that we seek to measure. This is in contrast to the ω term where the terms proportional to the FCA cross section are a small fraction of a large background signal (i.e. σ_FCAWN_AC/2m << 1). However, in practice, there is a large background signal that exists at 2ω as well, even in the absence of a sample. The background signal arises due to non-idealities in the experimental apparatus, such as harmonic distortion in the electro-optic-modulator that modulates the laser beam. This background component can be reduced by splitting the laser beam into a sample and reference beam and using the reference beam as feedback to reduce the background signal at 2ω in the sample beam [19]. Experimentally, this requires additional optical and electronic components, with accompanying complexity and introduction of systematic errors. By measuring the signal at ω instead, we find that it is unnecessary to split the laser and perform feedback. This simplifies the experimental apparatus and exemplifies the implementation of a truly single-beam technique. The signal that emerges from the detector at ω is given by:

\begin{matrix} S_{ω} (t) = ζ (ω) T P_{0} e^{- α W} [(1 - \frac{σ_{F C A} W}{2} N_{D C} - \frac{σ_{F C A} W}{2 m} N_{A C} \cos ϕ_{F C A}) m \cos ω t \\ - \frac{σ_{F C A} W}{2} N_{A C} \sin ϕ_{F C A} \sin ω t], \end{matrix}

where ζ(ω) is the transimpedance response of the detector (i.e. the conversion of optical power to voltage). In Eq. (7) we have only considered the terms at the frequency ω since DC terms and the terms at 2ω are rejected by the lock-in amplifier upon demodulation. Substituting in the expressions for the free-carrier population, N_DC and N_AC (Eqs. (3)–(4)) and using

\sin ϕ_{F C A} = ω τ / \sqrt{1 + ω^{2} τ^{2}}

and

\cos ϕ_{F C A} = 1 / \sqrt{1 + ω^{2} τ^{2}}

we find:

\begin{matrix} S_{ω} (t) = ζ (ω) T P_{0} e^{- α W} m [(1 - σ_{F C A} \frac{P_{0} λ f_{a} τ}{h c π w^{2}} - σ_{F C A} \frac{P_{0} λ f_{a} τ}{h c π w^{2}} \frac{1}{1 + ω^{2} τ^{2}}) \cos ω t \\ - σ_{F C A} \frac{P_{0} λ f_{a} τ}{h c π w^{2}} \frac{ω τ}{1 + ω^{2} τ^{2}} \sin ω t] . \end{matrix}

From Eq. (8) it is clear that the FCA components of the signal are quadratic in incident power P₀. Furthermore, it can be seen that these terms are related to the effective recombination lifetime τ, which is the parameter we seek to measure. To isolate the FCA component we first normalize out the prefactor in front of the brackets. This linearizes the signal with respect to power and removes the detector response from the signal. This is done by measuring a background spectrum S_b without a sample in place. When the sample is removed, the optical power into the detector is adjusted so that it matches the power when the sample was in place, neglecting the small contribution from the FCA term (σ_FCAP₀λf_aτ/hcπw² << 1), such that S_b ≈ζ(ω)TP₀e^–αWmcos ωt. Dividing Eq. (8) by the magnitude of S_b (i.e. ζ(ω)TP₀e^–αWm) results in the linearized signal s(t):

\begin{matrix} s (t) = (1 - σ_{F C A} \frac{P_{0} λ f_{a} τ}{h c π w^{2}} - σ_{F C A} \frac{P_{0} λ f_{a} τ}{h c π w^{2}} \frac{1}{1 + ω^{2} τ^{2}}) \cos ω t \\ - σ_{F C A} \frac{P_{0} λ f_{a} τ}{h c π w^{2}} \frac{ω τ}{1 + ω^{2} τ^{2}} \sin ω t . \end{matrix}

Note that the normalization procedure has removed the detector response, transmission factor, modulation depth and band-to-band attenuation factor from the signal.

Equation (9) shows that the signal detected at ω is due to a superposition of the laser modulation signal and a FCA signal. Figure 2 shows a visual representation of how these signals arise. In principle, information about the carrier lifetime can be determined by measuring the in-phase-component of the signal, the out-of-phase component of the signal or the phase of the signal via the ratio of the in- and out-of-phase components. In this work, the FCA contribution to the signal is a factor of 10⁻³ times the direct transmission of the laser beam. A consequence of this is that the FCA contribution to the in-phase component and to the phase of the signal is small and difficult to separate from the much larger background. However, from Eq. (9) it is clear that the out-of-phase component of the signal is directly proportional to the FCA term. Isolating this term by demodulation with sin ωt we arrive at the out-of-phase single-beam signal Y:

Y = σ_{F C A} \frac{P_{0} λ f_{a} τ}{h c π w^{2}} [\frac{ω τ}{1 + ω^{2} τ^{2}}] .

The dimensionless prefactor in front of the

ω

dependence may be written as:

Y_{a} = σ_{F C A} K P_{0},

where K is a constant given by:

K = \frac{λ f_{a} τ}{h c π w^{2}} .

From a measurement of the frequency dependence of Y, which peaks at ωτ = 1, we can determine τ. From a measurement of the prefactor of Y we can determine σ_FCA, using known constants and measurable parameters.

Fig. 2 Visual representation of single-beam self-induced absorption signal at different temporal points along a single excitation period. (a) Laser beam passing through wafer from top to bottom, with the AC and DC components represented by red and green shading, respectively. The AC component generates a free-carrier population at the same frequency with a phase lag of ϕ_FCA = tan^–1 ωτ, shown for the peak condition of Eq. (10) when ωτ = 1. The DC component is periodically attenuated by free-carrier absorption. Band-to-band absorption effects are not shown since they have been removed during signal normalization of Eq. (8). (b) Magnitude of the free-carrier population, n, as a function of time. (c) Decomposition of the signal, S, from the transmitted laser beam. The ‘MOD’ component is due to the AC modulation of the laser itself. The ‘FCA’ component is due to periodic free-carrier absorption of the DC component of the laser and is proportional to the free-carrier population and opposite in sign. Equation (9) includes the superposition of these signals.

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In this section we have developed the theory for the single-beam self-induced absorption technique by considering the specific case where the laser has a harmonic time dependence. There is no loss of generality in this solution since we can formulate the solution for an arbitrary time dependence by Fourier synthesis. Similarly, this technique could readily be extended to time domain measurements.

4. Results and discussion

A comparison of the single and dual-beam frequency spectra is shown in Fig. 3. Both frequency spectra are fitted to the frequency dependence in Eq. (10). There is excellent agreement between the modelled frequency dependence and the experimental data, for both single and dual beam data. The lifetimes extracted from the fits of the single and dual beam data are τ_SB = 122.7 $\pm$ 0.5 μs and τ_DB = 123.9 $\pm$ 0.7 μs, respectively. These values agree to within the precision of the fit, indicating that the single-beam self-induced absorption technique measures the same lifetime as dual-beam modulated pump/probe. This result is quite profound. It shows that a second, separate optical source is not required for these types of semiconductor pump/probe studies; a single laser beam can extract identical information by exploiting the non-linear optical absorption of the semiconductor that manifests from self-induced absorption. Typically the non-linear component of optical absorption in a semiconductor is ignored because it is much smaller than the band-to-band component. Indeed, according to the amplitude of the single-beam data in Fig. 3, the signal due to FCA is ~10⁻³ times the linear component of transmission, which is negligible for most applications. Non-linear transmission has been observed by other authors, but only for pulsed lasers where the average power is very high [16,17]. We believe this is the first study to observe the non-linearity at low power in a CW laser. This is made possible by the phase shift between the excitation and FCA signals. The signal component that is in-phase with the optical excitation contains contributions from both the excitation and FCA signals. The out-of-phase component is due to FCA alone, and this component can be isolated by the phase-sensitive detection of the lock-in amplifier.

Fig. 3 Comparison of single-beam self-induced absorption and dual-beam modulated pump/probe techniques. The single-beam data is given in absolute units whereas the dual-beam data is scaled arbitrarily for comparison. Symbols represent experimental datapoints and solid lines represent the best-fit to the frequency dependence in Eq. (10). The intensity of the pump laser illuminating the silicon wafer is 1.412 W/cm² for both single and dual-beam data sets.

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In addition to the lifetime τ, the fit to Eq. (10) also extracts the prefactor Y_a. Equation (11) predicts that the prefactor of the out-of-phase signal divided by K should scale linearly with incident power. This is a result of the normalization in Eq. (9) which linearizes the quadratic dependence of the FCA signal. The prediction of linearity is confirmed in Fig. 4 which shows a plot of Y_a/K vs P₀. This plot is generated by fitting Eq. (10) for Y_a and τ at various power levels, then using τ to compute K and forming the quotient Y_a/K. The slope is the FCA cross-section, which we find is σ_FCA = (3.95 $\pm$ 0.1) × 10⁻¹⁰ μm². The cross-section is a material constant and the value that we measure is comparable to values reported elsewhere in the literature. Svantesson measured [16] a cross section of σ_FCA = (5.1 $\pm$ 0.4) × 10⁻¹⁰ μm² at 1064 nm. One reason for the discrepancy between the measurements in this work and in the literature is that σ_FCA increases with increasing carrier concentration [20]. In Svantesson’s work [16] the cross section is an average cross section measured over a range of n = 10¹⁶–10¹⁹ cm^–3, whereas in this work the average (DC) carrier concentration was approximately n = 1 × 10¹⁶ cm^–3.

Fig. 4 Plot of Y_a/K for various incident powers. The plot is linear as expected, and yields a FCA cross section of σ_FCA = (3.95 ± 0.1) × 10⁻¹⁰ μm².

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The minimum-detectable lifetime of the single-beam self-induced absorption technique is limited either by the bandwidth of the instrumentation or the detection sensitivity of the signal amplitude. The electro-optic modulator used in this work has a bandwidth of 25 MHz leading to a minimum lifetime of 6.4 ns. Shorter lifetime detection can be achieved by using faster electro-optic modulators. It should be noted that since the signal strength is proportional to lifetime (Eqs. (11)–(12)), the minimum detectable lifetime is not necessarily limited by the bandwidth but by the detection limits of the instrumentation. According to Eq. (9) the signal consists of two components. The factor of unity corresponds to the signal due to the direct transmission of the laser through the semiconductor, whereas the other factors are due to FCA. As discussed above, this FCA component is ~1000 times smaller than the direct component and consequently the dynamic range of our measurement system is the limiting factor in our signal resolution. In order for the FCA signal to be digitized alongside the direct signal, the ADC on the lock-in amplifier must have a sufficient dynamic range. The lock-in amplifier used in this work has a 14-bit ADC and so we can estimate a detection limit of 2⁻¹⁴. Letting ν be the detection limit imposed by the ADC and using Eqs. (11)–(12), the minimum detectable lifetime is given by:

τ_{min} = \frac{h c ν}{I_{0} λ f_{a} σ_{F C A}},

where I₀ is the intensity of the laser. To formulate Eq. (13) we set the amplitude of the dimensionless signal Y_a equal to the detection limit ν. Using the experimental parameters of this work, the minimum detectable lifetime is ~3 μs. This can be improved by increasing the intensity of the laser, either by reducing the beam radius or increasing power. The dynamic range of the experiment could also be extended by using a higher resolution ADC or by using analog detection.

Single-beam pump/probe-like techniques have been demonstrated previously for studying photoreflectance in silicon [19,21]. In these studies, a harmonically modulated pump laser emitting 532 nm radiation is split into two beams, with one beam illuminating the sample and the second beam used for feedback. Feedback purifies the sample beam to reduce spurious harmonics away from the fundamental frequency. When the beam interacts with the sample, it modulates the refractive index at the modulation frequency ω inducing a periodic change in the reflectance of the sample. The reflected beam is collected by a photodetector for analysis. Since the laser beam and reflectance are varying at ω, frequency mixing leads to a signal at 2ω which is due purely to photoreflectance. In our work, the modulation of the signal arises due to modulation of the absorption coefficient of the material (i.e. the imaginary part of the refractive index), whereas in photoreflectance work it arises due to modulation of the reflection coefficient (i.e. the real part of the refractive index). Though we have neglected it in our analysis, it is expected that we should also observe the effect of photoreflectance since a change in reflection results in a change in transmission. By calculating the change in the real part of the refractive index, based on literature values [22], we find that the photoreflectance is about 7 × 10⁻⁴ times the change in transmittance due to FCA. Thus we are justified in neglecting photoreflectance from our mathematical analysis. Even in the case where photoreflectance is expected to be comparable to or greater than the FCA signal, its contribution could be removed by illuminating the sample at Brewster’s angle with p-polarized light.

Another well-known single-beam method for studying non-linear optical effects is the Z-scan technique. In Z-scan, the position of the sample under study is swept through the focal point of a strongly focused laser. By measuring the transmitted signal as a function of position, the non-linear refractive index and absorption coefficient may be determined [23,24]. The signal in the present work arises due to self-induced absorption, which is non-linear with optical power. In principle, the technique could be used to measure the nonlinear absorption coefficient related to self-induced absorption. From Eq. (9) we find that at DC the linearized signal becomes s = 1 – 2σ_FCAλI₀f_aτ/hc. With appropriate calibration, a measurement could be used to determine τ if σ_FCA is known, or σ_FCA if τ is known. This approach is implemented in quasi-steady state free carrier absorption (QSS-FCA) measurements [25], with an intentionally unfocussed laser beam. When the width of the focused laser beam becomes comparable to or less than the diffusion length of free-carriers, then the single-beam self-induced absorption signal will become sensitive to diffusion of free-carriers out of the volume illuminated by the laser beam. In this work, we have also intentionally used an expanded laser beam to avoid diffusion so that the signal is only sensitive to recombination. From this perspective, the Z-scan experimental approach provides no advantages for measuring carrier lifetime in Silicon. Furthermore, the modulation methods described above provide a direct measurement of carrier lifetime without the need for determining σ_FCA.

Implicit in the derivation of the self-induced absorption signal is that the band-to-band and FCA processes are independent, i.e. the injection rate of free-carriers due to band-to-band absorption of the laser is unperturbed by the rate of FCA. This condition is achieved when $α_{F C A}^{A C} < < α$ , where $α_{F C A}^{A C}$ is the AC component of FCA due to the periodic injection of carriers. We estimate that in this work the maximum carrier density that is injected is n_AC = 1 × 10¹⁶ cm^–3, which given the FCA cross section that we measured (σ_FCA = (3.95 $\pm$ 0.1) × 10⁻¹⁰ μm²) results in α_FCA = 3.95 × 10⁻² cm^–1. In this work we use a 1064 nm laser, which has an absorption coefficient of α = 9.85 cm^–1 and so the criterion $α_{F C A}^{A C} < < α$ is satisfied. The criterion only applies to the AC component of α_FCA since our signal is only sensitive to the AC modulation of free-carriers. Indeed, the DC contribution to α_FCA from background dopants may be comparable to or even exceed α. In this case, the rate of free-carrier injection is reduced since band-to-band absorption is competing with FCA, leading to a reduced signal strength. Excessively heavy doping will reduce the rate of band-to-band absorption to the point where the linearized signal strength drops below the detection limit of the lock-in amplifier. The lock-in amplifier used in this study has a 14 bit ADC so we estimate a detection limit of 2⁻¹⁴. According to the single-beam data in Fig. 3 the linearized signal strength is ~10⁻³ so we could tolerate ~16 times reduction in free-carrier injection rate before the signal drops below the detection limit. Assuming then that α/(α + α_FCA) = 1/16 and using the σ_FCA value obtained in this work we estimate that the doping density would need to be as high as n = 4 × 10¹⁹ cm^–3 in order for the signal to drop below the detection limits of the lock-in amplifier. Since the signal strength depends on several factors such as lifetime, beam radius, and incident power (see Eq. (10)), the particular doping density that can be tolerated will depend on the experimental conditions.

The non-linear dependence of the FCA signal that is observed in this work arises due to mixing from two linear processes. The generation of free-carriers and the attenuation of the laser due to those free-carriers both vary linearly as a function of P₀ which leads to a P₀² dependence. In addition to this non-linearity there are more traditional non-linear processes that occur in semiconductors such as two-photon absorption (TPA) [26]. The magnitude of TPA is proportional to the intensity of the laser. Since the laser that is used in this work is defocused and CW we do not expect to observe TPA. According to [27] the TPA coefficient for silicon at 1064 nm is β = 1.5 cm/GW. Using the laser intensity 1.412 W/cm² leads to α_TPA = 2.2 × 10⁻⁹ cm^–1 which is about 7 orders of magnitude smaller than the FCA coefficient observed in this work and can be safely neglected.

5. Conclusions

The experimental results of Fig. 3 and Fig. 4 demonstrate proof-of-concept for the single-beam self-induced absorption technique and quantitatively validate the mathematical model that has been derived in this work. These results are quite remarkable because it has been shown that a second laser beam is not required in order to extract information acquired from semiconductor modulated pump/probe measurements. This is a dramatic simplification which opens up the possibility of incorporating pump/probe studies into applications that would otherwise be impractical with two beams. The most natural application for the single-beam self-induced absorption technique is for monitoring of semiconductor processing in situ. For example, ellipsometry [28,29], x-ray diffraction [30,31], electron-microscopy [32,33], and Fourier Transform Infrared (FTIR) Spectroscopy [34,35] have all been used to study film growth in situ in Chemical Vapor Deposition (CVD) or Atomic Layer Deposition (ALD) reaction chambers. What these techniques all have in common is that they study the properties of the film being grown, and not their effect on the substrate. The single-beam self-induced absorption technique is uniquely suited to studying the underlying substrate. In photovoltaic applications, CVD dielectric coatings grown on the semiconductor surface reduce the rate of recombination at the surface by soaking up dangling bonds and/or by inducing a surface field that repels minority carriers, which increases the recombination lifetime and improves device performance [36]. Device manufacturers are interested in optimizing the film growth parameters in order to maximize the recombination lifetime. For plasma-assisted CVD, parameters such as RF power, gas flow rate, gas flow ratios, deposition temperature, and deposition pressure can all affect the final lifetime of the wafer [37]. Optimization of the growth parameters currently requires measuring the lifetime ex situ, which consumes time and resources. With the single-beam self-induced absorption technique, the lifetime could be measured in situ during the film-growth process and thus the quality of the film could be measured in real-time and used as direct feedback for growth process optimization. Measurements in situ using a dual-beam approach would be impractical since two lasers have to be simultaneously overlapped inside a CVD chamber. It would be extremely challenging to ensure this overlap without direct access to the sample stage, which is atypical for large-scale CVD chambers. As well, dispersion effects due to heating of the windows of the CVD chamber and the gases that flow into it would cause a misalignment of the pump and probe beams during the deposition. With a single-beam self-induced absorption measurement, there is no concern about overlapping of pump and probe beams, which greatly simplifies the incorporation of a single-beam self-induced absorption for in situ monitoring. The single-beam self-induced absorption technique also has many interesting potential spectroscopic applications including and beyond semiconductor applications. As long as the absorption coefficient and sample thickness can be arranged so that αW ~1 and the primary absorption process is dominant, the wavelength could be tuned to obtain spectrally-resolved information.

Appendix Excess carrier density in a semiconductor

When a semiconductor is illuminated with light that has energy above its bandgap, free electron-hole pairs (free-carriers) are excited due to band-to-band absorption of the light. For a pulsed excitation where the pulse-width is much shorter than the characteristic decay time of free-carriers, the average carrier density as a function of time [38] is given by:

n_{i} (t) = \frac{8 ϕ_{0} α f_{a} e^{- \frac{α W}{2}}}{W (1 - e^{- α W})} \sum_{k} A_{k} e^{- \frac{t}{τ_{k}}},

where f_a is the fraction of optical power absorbed in the wafer, ϕ₀ is the number of photons per unit area for the pulse duration, W is the wafer thickness, τ_k is the series effective lifetime. The series effective lifetime and the coefficient A_k are given by:

τ_{k} = {(τ_{b}^{- 1} + α_{k}^{2} D)}^{- 1},

\begin{matrix} A_{k} = \frac{\sin (\frac{α_{k} W}{2})}{(α^{2} + α_{k}^{2}) (α_{k} W + \sin α_{k} W)} \times \\ [α \sinh (\frac{α W}{2}) \cos (\frac{α_{k} W}{2}) + α_{k} \cos h (\frac{α W}{2}) \sin (\frac{α_{k} W}{2})] . \end{matrix}

Here α_k = 2γ_k/W, where γ_k are roots to the transcendental equation cot γ_k = (2D/SW)γ_k, D is the ambipolar diffusion coefficient, τ_b is the bulk recombination lifetime and S is the surface recombination velocity. Equation (14) is the impulse-response of the excess carrier density when the optical generation rate of free-carriers is g(t) = δ(t), where δ is the Dirac delta function.

For an arbitrary generation rate g(t) the excess carrier density at time t is given by the convolution of n_i(t) and g(t):

n (t) = \int_{0}^{t} n_{i} (t - t^{'}) g (t^{'}) d t^{'} .

In this study the generation rate is a harmonic function of time with a DC offset, so g(t) = 1 + me^iωt where m is the modulation depth, and ω is the angular modulation frequency. Substituting this into Eq. (17), and taking the limit where t >> τ, the harmonically varying excess carrier density is given by:

\tilde{n} (t) = \frac{8 ϕ α f_{a} e^{- \frac{α W}{2}}}{W (1 - e^{- α W})} \sum_{k} A_{k} τ_{k} (1 + m \frac{1}{1 + i ω τ_{k}} e^{i ω t}) .

The tilde on the symbol n denotes that the carrier density is a complex number. To find the true carrier density, the real part of ñ(t) is taken. The factor ϕ is the number of photons per unit area, per unit time. For a Gaussian laser beam, this factor is ϕ = exp(–2r²/w²)2P₀λ/hcπw², where P₀ is the total incident power of the laser, λ is the wavelength, w is the radius of the Gaussian beam, h is Planck’s constant, and c is the speed of light in vacuum. The radius is defined by the point where the intensity is 1/e² of its peak value. The free-carrier population takes on the same radial dependence as the laser since the beam radius is assumed to be much larger than the diffusion length of free-carriers. The coordinate r describes the distance from the center of the beam. Substituting ϕ into Eq. (18):

\tilde{n} (r, t) = \frac{16 P_{0} λ α f_{a} e^{- \frac{α W}{2}}}{h c π w^{2} W (1 - e^{- α W})} e^{- \frac{2 r^{2}}{w^{2}}} \sum_{k} A_{k} τ_{k} (1 + m \frac{1}{1 + i ω τ_{k}} e^{i ω t}) .

Equation (19) is a general description of the free-carrier density due to a harmonically varying optical excitation. The constants τ_k are the characteristic decay constants of the normal modes of the impulse response n_i, where k is the index of the mode. The decay constants τ_k have contributions from the bulk recombination lifetime τ_b and from the surface recombination velocity S via the term α_k. When the surface recombination velocity is sufficiently small that 2SW/D << 1, the first transcendental root can be approximated as α₁² ≈ 2S/DW. This is the case of interest for wafers used to fabricate high efficiency solar cells. With this approximation, A_k = sinh(αW/2)/4α for k = 1 and A_k = 0 for k ≠ 1. Substituting A_k into Eq. (19) and dropping the subscript ‘k’ from τ_k we arrive at a simplified version of the general case for ñ(r,t):

\tilde{n} (r, t) = \frac{2 P_{0} λ f_{a} τ}{h c π w^{2} W} e^{- \frac{2 r^{2}}{w^{2}}} (1 + m \frac{1}{1 + i ω τ} e^{i ω t}),

where τ is the effective recombination lifetime, which in the limit of 2SW/D << 1 is given by:

τ = {(τ_{b}^{- 1} + 2 S / W)}^{- 1} .

Taking the real part of Eq. (20) yields the carrier density n(r,t):

n (r, t) = \frac{2 P_{0} λ f_{a} τ}{h c π w^{2} W} e^{- \frac{2 r^{2}}{w^{2}}} (1 + \frac{m}{1 + ω^{2} τ^{2}} (\cos ω t + ω τ \sin ω t)) .

This equation can be rewritten more succinctly with a single trigonometric function:

n (r, t) = \frac{2 P_{0} λ f_{a} τ}{h c π w^{2} W} e^{- \frac{2 r^{2}}{w^{2}}} (1 + \frac{m}{\sqrt{1 + ω^{2} τ^{2}}} \cos (ω t - ϕ_{F C A})),

where ϕ_FCA is the phase of the free-carrier population with respect to the drive signal:

ϕ_{F C A} = \tan^{- 1} ω τ .

Finally we can write Eq. (23) in terms of a DC and AC component:

n (r, t) = e^{- \frac{2 r^{2}}{w^{2}}} (N_{D C} + N_{A C} \cos (ω t - ϕ_{F C A})),

where N_DC and N_AC are given by:

N_{D C} = \frac{2 P_{0} λ f_{a} τ}{h c π w^{2} W},

N_{A C} = \frac{2 P_{0} λ f_{a} τ}{h c π w^{2} W} \frac{m}{\sqrt{1 + ω^{2} τ^{2}}} .

Funding

Natural Sciences and Engineering Research Council of Canada (NSERC) (03736); Canada Foundation for Innovation (CFI) (32168).

Acknowledgements

We thank Harold Haugen for a valuable conversation on this topic.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Single-beam lifetime measurements via self-induced optical absorption

Abstract

1. Introduction

2. Experimental

3. Theory

4. Results and discussion

5. Conclusions

Appendix Excess carrier density in a semiconductor

Funding

Acknowledgements

Disclosures

References

Cited By

Figures (4)

Equations (27)

Optics Express