Determination of excitation profile and dielectric function spatial nonuniformity in porous silicon by using WKB approach

Wei He; Igor V. Yurkevich; Leigh T. Canham; Armando Loni; Andrey Kaplan

doi:10.1364/OE.22.027123

1. Introduction

Investigation of the nano-porous silicon (PS) has attracted tremendous fundamental research activity and investment into development of applications. The success of the PS is attributed to a variety of factors, most crucial among them being those related to the wide use of silicon in microelectronics, the well controlled and reproducible dimensions of PS owing to a robust fabrication technology, and the ability to produce quantities on an industrial scale. The potential range of applications spreads over research areas and development activities including the drug and food industry [1], optical sensing and wave-guiding [2], ion exchange, molecular isolation and purification [3], photonics devices [4,5], antireflection coatings in solar cells [6], and biosensing [7] and many others. Successful development of applications, in the future, requires reliable and verified models describing the optical and conductive properties of the PS [8, 9].

In this investigation, we focused on measurements and modeling of the complex refractive index change induced by an external perturbation. In our work, the perturbation was applied in the form of free-charge-carriers excitation by a femtosecond laser pulse. However, the method we present here to investigate a spatially nonuniform complex refractive index can be applied to a broader range of applications, such as refractive index change in biosensing [7], nonuniform porosity and pore diameter in solar cells and detectors [10] and sensing material nonuniformity in coatings [11].

For the investigation we used PS membranes manufactured by the electrochemical etching method [12], which can produce a sponge-like network with nanometer scale pores (≪100 nm) and nanometer-scale silicon wires with diameter ≪ 50 nm. This nanometer-scale silicon has a number of unique features, producing a dramatic effect on optical and electrical compared to those of bulk silicon: a large surface area to volume ratio, a pore size smaller than the optical wavelength, a modified energy gap, and a constricted free-carrier mean free path. It has also been reported that a significant response can be observed in pump-probe measurements, and this technique was widely used in the previous works to study the excited state of the PS [13–15]. It was shown that the excitation of the free carriers can be described sufficiently well by the Drude response.

To obtain the experimental results we used pump–probe femtosecond pulses with identical spectrum spanning through the region between 770 and 820 nm. All the measurements were done at a 5 – ps time delay between the pump and probe, recording simultaneously the changes of transmittance and reflectance induced by the pump. The analysis focused on the retrieving of the complex dielectric function alteration caused by the pump excitation. We show that the assumption of uniform excitation and uniform dielectric function alteration is insufficient for constructing an optical model able to successfully describe the observed results. Therefore, we developed an analytical approach that allows us to find a nonuniformity of the excitation. This method is based on the Wentzel-Kramers-Brillouin approach (WKB) and allows us to find a smooth change of the dielectric function between the membrane facets. Using this method, we retrieved the depth–dependent change of the complex dielectric function induced by the pump. The optical model developed in this study can be used for more complex analysis of optical pump–probe measurements, such as determination of the excitation lifetime, reconstruction of carrier diffusion, and establishment high-frequency conductivity.

2. Experiment

2.1. Pump probe setup

The optical properties of optically excited PS membranes were investigated by using the traditional optical pump–probe method to measure induced reflectivity and transmission change at a 5 – ps delay time after pump excitation. A relatively long delay time was set to allow the nascent, hot, excited charge carriers to cool and relax to the bottom of the band edge, the delay time was sufficiently short to allow the recombination process to be neglected as it proceeds on a much longer time scale. For the measurements, a Coherent ultrafast laser system was used to provide ultrashort laser pulses. The system delivers ∼ 50 fs pulses of nearly Gaussian shape, covering the spectral range between 760 and 830 nm. The source beam was split by a pellicle beam splitter into intensive pump pulses and weak probe pulses in a ratio of ∼ 100 : 1. With the help of a computer-controlled retroreflector, the optical path length of the probe pulse can be precisely delayed with respect to that of the pump pulse, generating a difference between arrival times of the pulses on the surface of the sample. The pump fluence was adjusted by using an attenuator based on a combination of a rotatable half–wave plate and a near–Brewster angle reflection. Half–wave plate polarizers were used to set the probe and pump beams to the s– and p–polarization states, respectively, to avoid interference. The incident angle of the probe beam was set to 45°. The difference between the pump and the probe beam incident angles was 15°. The probe and pump were focused to ∼ 50 and ∼ 500 μm spot diameters, respectively, on the surface of the PS membrane. The spatial overlap between the pump and the probe was checked by a CCD camera equipped with a magnifying lens. The temporal overlap between the pump and probe pulses was determined by a second harmonic generation from a BBO crystal positioned at the place of the sample. For measurements of the transmission and reflection of the probe beam, the same spectrometer (Ocean Optics QE65 Pro) was used. Further information about the experimental setup can be found elsewhere [16–18]. The measurements presented here are in the form of a fractional change of the reflectance and transmittance: ΔT/T₀ = (T_t − T₀)/T₀ and ΔR/R₀ = (R_t − R₀)/R₀, where T_t and R_t are the transmittance and reflectance respectively, of the excited sample at a time delay t after the pump excitation; T₀ and R₀ are the transmittance and reflectance of the sample without excitation.

2.2. Sample preparation and characterization

The porous silicon layer was fabricated by electrochemical anodization of the surface of a 3″ diameter (100) silicon wafer (B-doped, 5 – 15 mΩcm), using an electrolyte comprised of methanol and 40% HF in a ratio 1:1; a current density of 30 mA/cm² and anodization time of 11 min was chosen to yield a layer with > 60% porosity and 11μm depth (calculated using a gravimetric calibration curve); the layer was detached from the underlying substrate, after anodization, by applying a 120 mA/cm² pulse (10 s) before removed from the electrolyte; the free-standing membrane was then rinsed in methanol and air-dried.

On the other hand, the porosity of > 50% and the thickness of ∼ 13μm of the PS membrane were estimated from the images of the cross section obtained by SEM (see Figure 1(a)–(c)).

Fig. 1 SEM images of the PS membrane sample, showing (a) the surface of the sample, and (b) and (c) cross sections of the sample

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Our main goal is to investigate the change of the dielectric function induced by an external perturbation. For this purpose, a well-founded model describing the optical response at the ground state, i.e., without application of the pump pulse, is needed. The model requires input information about the sample composition, porosity, and thickness and the dielectric function in the absence of the excitation.

To evaluate the dielectric function in the region of interest between 760 and 810 nm, we measured transmittance T₀, and reflectance R₀, of the membrane using the probe beam. The results are shown in Fig. 2. To analyse these measurements, we used a procedure based on the methodology and data provided elsewhere [19–24]. In brief, to construct the optical model describing the optical response of the PS membrane we used Forouhi-Bloomer model combined with the Bruggeman effective medium approximation.

Fig. 2 (a) Transmittance, T₀ and (b) reflectance R₀, of the PS membranes. The black dots show the experimental data; the red solid lines correspond to the best fitting.

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The four-term Forouhi-Bloomer interband model [25] was used to simulate the complex index of refraction of the nanocrystalline silicon constituent of the membrane:

n (E) = \sqrt{ε_{\infty}} + \sum_{i = 1}^{q} \frac{B_{0_{i}} \cdot E + C_{0_{i}}}{E^{2} - B_{i} \cdot E + C_{i}},

k (E) = {\begin{matrix} \sum_{i = 1}^{q} \frac{A_{i} \cdot {(E - E_{g})}^{2}}{E^{2} - B_{i} \cdot E + C_{i}} & E > E_{g} \\ 0 & E \leq E_{g} \end{matrix}

where

\begin{array}{l} B_{0_{i}} = \frac{A_{i}}{Q_{i}} \cdot (- \frac{B_{i}^{2}}{2} + E_{g} \cdot B_{i} - E_{g}^{2} + C_{i}), \\ C_{0_{i}} = \frac{A_{i}}{Q_{i}} \cdot [(E_{g}^{2} + C_{i}) \cdot \frac{B_{i}}{2} - 2 \cdot E_{g} \cdot C_{i}], \\ Q_{i} = \frac{1}{2} \cdot \sqrt{4 \cdot C_{i} - B_{i}^{2}} . \end{array}

and where n and k are the real and imaginary parts, respectively. (The dielectric function can be calculated by using the definition ε_c–Si = (n − ik)², where ε_c–Si assigns the dielectric function of the crystalline silicon.) This model is based on the crystalline silicon energy band structure and satisfies the Kramers–Kronig relation, where E_g is the energy band gap and E is the probing photon energy, ε_∞ is the dielectric constant corresponding to E → ∞, and A_i, B_i, and C_i are the phenomenological parameters related to the optical transitions around the critical points where the conduction and valence bands are approximately parallel to each other and the optical absorption is enhanced. Generally, four dominant critical point transitions can sufficiently describe the c–Si optical property [25]. Thus, four terms, q = 4, can be used to describe the optical property of crystalline silicon (c–Si) over a wide spectral range. A similar approach was also used in the previous works to simulate nanocrystalline silicon materials [26, 27].

Then, the Bruggeman effective medium approximation [28] is used to mix the phase of nano– c–Si with voids to express the effective dielectric function, ε_eff, of the PS membrane, which is given as

\begin{array}{l} ε_{eff} & = \frac{1}{4} {ε_{v} (3 p - 1) + ε_{c - Si} (2 - 3 p) \\ + \sqrt{{[ε_{v} (3 p - 1) + ε_{c - Si} (2 - 3 p)]}^{2} + 8 ε_{c - Si} ε_{v}}} . \end{array}

where ε_v = 1 is the dielectric constant of voids and p is the porosity of the PS membrane. The effective dielectric constant ε_eff was used in the transfer matrix method [18, 29] to calculate

T_{0}^{cal}

and

R_{0}^{cal}

, denoting the modeled transmittance and reflectance, respectively. We note that ε_eff was assumed to be uniform across the membrane depth (an assumption that holds for membranes thinner than 30 – 50 μm). The fitting parameters in the optical model were iteratively adjusted to minimize the difference between the calculated and measured values to obtain the best fitting shown in Fig. 2. The best fitting is obtained by setting the energy band gap, E_g, of nano–c–Si at 1.3ev; the composition and thickness used in the calculation are very similar to those obtained experimentally (a porosity is 54.5% and a thickness of 13.2μm). The phenomenological parameters, A_i, B_i, and C_i, are close to the values published in the original paper for the crystalline silicon [25]. The values of ε_c–Si and ε_eff obtained in the fitting process for the whole PS membrane are shown in Fig 3. We note that the dielectric function of the nano–crystalline constituent is slightly different from that of bulk silicon owing the presence of native silicon oxide and nanoscopic dimensions affecting the band structure of silicon crystallites [30, 31]. We also note that the absolute change of the values of ε_c–Si over the observed spectrum is greater than that of the corresponding effective dielectric functions ε_eff. This effect is owed to the dilution of the crystalline silicon phase and decreased contribution to the optical response. However, the relative change is the same for both functions.

Fig. 3 Real and imaginary parts of the dielectric functions of the crystalline silicon constituent (c–Si) of the PS membrane (solid lines) and the effective medium approximation (dashed lines). The left axis corresponds to the real parts; the right axis corresponds to the imaginary parts.

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3. Results and discussion

To investigate the optical properties of the PS membranes excited by the pump pulse the delay time was fixed at 5 ps and the fluence was set to 1.5 mJ/cm². Figure 4 shows the experimental results of the transient pump–probe fractional change of the transmission and reflection, ΔT/T₀ and ΔR/R₀, respectively, as a function of the probing wavelength.

Fig. 4 Transient pump–probe transmission and reflection, ΔT/T₀ and ΔR/R₀, measured on the PS membranes at a delay time of 5 ps. The black dots represent the experimental results. The red solid lines show the fitting results using the uniform optical model. The bottom plots show the change to the dielectric function induced by the pump and used for the calculation of ΔT/T₀ and ΔR/R₀. (a) A uniform model provides good fitting to ΔT/T₀, but shows a discrepancy in the fitting to ΔR/R₀; (b) a uniform model fits well ΔR/R₀, but fails to describe ΔT/T₀.

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3.1. Uniform excitation model

We attempted to simulate the experimental pump–probe results by assuming a uniform change of the complex dielectric function across the membrane depth induced by the pump. The dielectric function of the nano-silicon fraction excited by the pump, ε_Si–pump, can be described as a sum of the dielectric functions at the ground state (shown in Fig. 3), ε_c–Si, and the response of the free carrier, Δε_fcr, excited by the pump:

ε_{Si - pump} = ε_{c - Si} + Δ ε_{fcr} .

In general, there are additional possible physical effects expected to contribute to the change of the linear optical properties of an optically excited semiconductor, such as band filling, band structure renormalization and phonon excitation, but in our experimental conditions these effects are negligible [32].

The Bruggeman model was then implemented again to express the effective dielectric function of the excited PS membrane and the transfer matrix method was used to evaluate the change of the transmittance and reflectance, ΔT/T₀ and ΔR/R₀, respectively, induced by the excitation. However, we were unable to fit simultaneously ΔT/T₀ and ΔR/R₀. Figure 4(a) shows the cases where a uniform change of the dielectric function provides a good fitting of ΔT/T₀, but was not satisfactory for ΔR/R₀. In contrast, on Fig. 4(b) ΔR/R₀ was reasonably well-fitted, but ΔT/T₀ was not. Any attempts to simultaneously fit the change in reflectance and transmittance resulted in discrepancies between the experiment and calculations. We concluded that no unique solution exists for a uniform Δε_fcr that can describe the optical response of the optically excited PS membrane. This conclusion is well-founded on a simple argument that, in the first instance after the excitation, the spatial profile of the excited carriers reproduces the absorption profile of the pump and, thus, it is not uniform because the pump intensity drops along the propagation path through the membrane. Therefore, we developed a new procedure to analyze the results and calculate the variation of the optical properties as a function of the membrane depth.

3.2. Nonuniform model

The model we used to describe the optical response of the optically excited membrane is based on the WKB method, in which a smooth nonuniform change of the dielectric properties as a function of the depth is assumed [33]. In our case, this change is related to the alteration of the density of excited charge carriers as a function of the membrane depth along the coordinate z. Therefore, the dielectric function of the membrane is given by ε′_Si–pump(z) = ε_c–Si + Δε_fcr(z). This function was used in the Bruggeman model to obtain the effective dielectric function ε′_eff (z) and calculate ΔT/T₀ and ΔR/R₀ by fitting the experimental results. Hence, in this fitting procedure Δε_fcr(z) can be retrieved. We note that, in general, one may expect nontrivial oscillations as a function of the angle of diffusively scattered light [34], but here we will concentrate on the specular reflection and transmission only, which are well described by a model with a dielectric function depending exclusively on the perpendicular coordinate.

3.2.1. Definition of boundary characteristic matrix

First, we derive the boundary transfer matrix method that is used to calculate the transmittance and reflectance of the membrane. Figure 5(a) shows a boundary between media i and j. The complex amplitudes of light incoming from the left and right are expressed as $a_{i}^{in}$ and $a_{j}^{in}$ , respectively, while the outgoing amplitudes are given by $a_{i}^{out}$ and $a_{j}^{out}$ . Neglecting surface roughness, which causes cross–polarized scattering [35], we may define a characteristic (transfer) boundary matrix T̂_ij for s-polarization only that relates the fields on both sides of the boundary, i and j:

[\begin{matrix} a_{i}^{in} \\ a_{i}^{out} \end{matrix}] = {\hat{T}}_{i j} \cdot [\begin{matrix} a_{j}^{out} \\ a_{j}^{in} \end{matrix}] = [\begin{matrix} T_{i j}^{11} & T_{i j}^{12} \\ T_{i j}^{21} & T_{i j}^{22} \end{matrix}] \cdot [\begin{matrix} a_{j}^{out} \\ a_{j}^{in} \end{matrix}] .

Fig. 5 (a) Complex amplitudes of the electric fields of light crossing a boundary; amplitude coefficients for light incoming in the forward (b) and reverse (c) directions.

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To obtain the elements of the matrix T̂_ij, the two special cases shown in Figs. 5(b) and 5(c) are considered. Figure 5(b) depicts the case of light with a unitary field amplitude crossing the boundary from i to j. The Equation 5 for such a case can be presented as

[\begin{matrix} 1 \\ r_{i j} \end{matrix}] = {\hat{T}}_{i j} \cdot [\begin{matrix} t_{i j} \\ 0 \end{matrix}],

where r_ij and t_ij are the amplitudes of reflected and transmitted light, respectively. Similarly, for the field crossing the boundary from j to i, the expression can be formulated as

[\begin{matrix} 0 \\ t_{j i} \end{matrix}] = {\hat{T}}_{i j} \cdot [\begin{matrix} r_{j i} \\ 1 \end{matrix}] .

The elements of the matrix T̂_ij can be derived by using Eq 6 and Eq 7:

{\hat{T}}_{i j} = [\begin{matrix} \frac{1}{t_{i j}} & - \frac{r_{j i}}{t_{i j}} \\ \frac{r_{i j}}{t_{i j}} & t_{j i} - \frac{r_{i j} r_{j i}}{t_{i j}} \end{matrix}] .

Since r_ij = −r_ji and t_ij = 1 + r_ij, the Eq 8 can be simplified as the following:

{\hat{T}}_{i j} = \frac{1}{t_{i j}} \cdot [\begin{matrix} 1 & r_{i j} \\ r_{i j} & 1 \end{matrix}] .

We note that, T̂_ij⁻¹ = T̂_ji.

3.2.2. WKB approximation for light propagating through a medium with a spatially nonuniform dielectric function

Using the boundary transfer matrix developed above, we will derive a characteristic matrix based on the WKB method to calculate the transmittance and reflectance of light propagating through a membrane with a spatially nonuniform optical response.

Figure 6 shows a cross-section of a membrane of the thickness d. There are three regions denoted as ①, ② and ③. The interfaces between the regions ① and ②, and between ② and ③ can be described by the boundary transfer matrices T̂₁₂ and T̂₂₃, respectively. According to Eq. 8, these matrices can be presented as the following:

{\hat{T}}_{12} = \frac{1}{t_{12}} [\begin{matrix} 1 & r_{12} \\ r_{12} & 1 \end{matrix}] {\hat{T}}_{23} = \frac{1}{t_{23}} [\begin{matrix} 1 & r_{23} \\ r_{23} & 1 \end{matrix}]

Fig. 6 Schematic representation of a membrane cross-section. Regions ① and ③ represent the membrane interfaces; ② denotes the membrane region with a nonuniform dielectric function.

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To derive the matrix, T̂_m, describing the optical response of s-polarized light propagating between the interfaces, we assign the electric field of light as E(z) = E₊(z) + E₋(z), where E₊(z) and E₋(z) represent the fields of light propagating from left to right and in the reverse directions, respectively. k_i and k_j are the wave vectors of light outside and inside the membrane. The electric field E_±(z) at any depth position z is given by the following:

E_{\pm} (z) = \frac{a_{\pm}}{\sqrt{q (z)}} \cdot e^{\pm i \int_{0}^{z} d z^{'} q (z^{'})},

where

q (z) = \sqrt{\frac{ω^{2}}{c^{2}} ε_{eff}^{'} (z) - k_{x}^{2}} .

Here $k_{x} = \frac{ω}{c} \sin θ$ defines the tangential component of the wave vector, and θ is the incident angle of the probing beam. The matrix T̂_m relating the fields between the interfaces inside the membrane, can be expressed as the following:

[\begin{matrix} E_{+} (0) \\ E_{-} (0) \end{matrix}] = {\hat{T}}_{m} [\begin{matrix} E_{+} (d) \\ E_{-} (d) \end{matrix}] .

Using Eq. 11 we define the phase as

ψ (z, λ, θ) = \int_{0}^{d} d z q (z) = \frac{ω}{c} \int_{0}^{d} d z \sqrt{ε_{eff}^{'} (z, λ) - \sin^{2} θ}

. Thus, the matrix T̂_m can be presented in the following way:

{\hat{T}}_{m} = \sqrt{\frac{q (d)}{q (0)}} [\begin{matrix} e^{- i ψ} & 0 \\ 0 & e^{i ψ} \end{matrix}]

with the phase ψ being a function of the depth z and the parameters λ and θ.

Finally, the characteristic matrix for the entire system can be expressed as a product of the individual matrices: T̂₁₂, T̂_m and T̂₂₃:

\begin{array}{l} \hat{T} = {\hat{T}}_{12} \cdot {\hat{T}}_{m} \cdot {\hat{T}}_{23} = \frac{1}{t_{12} t_{23}} \sqrt{\frac{q (L)}{q (0)}} \cdot \\ [\begin{array}{l} e^{- i ψ} + r_{12} r_{23} e^{i ψ} & e^{i ψ} r_{12} + e^{- i ψ} r_{23} \\ e^{- i ψ} r_{12} + e^{i ψ} r_{23} & e^{i ψ} + r_{12} r_{23} e^{- i ψ} \end{array}] . \end{array}

According to Eq. 9, the reflection and transmission coefficients are given by

r_{i j} = \frac{T_{i j}^{21}}{T_{i j}^{11}}

and

t_{i j} = \frac{1}{T_{i j}^{11}}

. Thus, by combing Eq. 14 and the Fresnel equations, the reflection and transmission coefficients of the PS membrane are given by

r = \frac{r_{0} e^{- i ψ} - r_{d} e^{i ψ}}{e^{- i ψ} - r_{0} r_{d} e^{i ψ}}, t = \frac{1}{\sqrt{\frac{q (d)}{q (0)}}} \cdot \frac{(1 + r_{0}) (1 - r_{d})}{e^{- i ψ} - r_{0} r_{d} e^{i ψ}} .,

where r₀ and r_d are given by:

r_{0} = \frac{k_{z} - q (0)}{k_{z} + q (0)}, r_{d} = \frac{k_{z} - q (d)}{k_{z} + q (d)},

where

k_{z} = \frac{ω}{c} \cos θ .

To calculate the reflectance and transmittance one can use the definitions: R_t = |r|² and T_t = |t|².

One can see that the only unknown in our experiment is a complex function Δε_fcr(z), which can be reasonably well retrieved from a simultaneous fitting of the reflectance and transmittance.

3.2.3. Results of fitting and determination of Δε_fcr(z)

To simplify the fitting process, the fitting parameters were defined as Δε_fcr(0, λ) and Δε_fcr(d, λ) for the front surface and rear surfaces of the PS membrane, respectively, and a smooth decaying function joining these points. The fitting parameters and function are located in the complex plane; their real and imaginary parts are related by the Kramers-Kronig bidirectional relations. By using these parameters the front and rear surface reflection coefficients r₀ and r_d and the accumulated phase ψ can be estimated and iteratively adjusted minimising the difference between the calculated and the measurement results of ΔT/T₀ and ΔR/R₀. Figure 7 shows the best fitting results for the optical model with a nonuniform complex dielectric function. The results of the fitting represented by the red solid lines reproduce well the measurements denoted by the black dots. It is clearly seen that the nonuniform model provides better fitting to the results than the uniform one. We note here that for the fitting we used the multiple linear regression algorithm which provided the fitting error of χ = 0.0016. On the other hand the fitting errors for the uniform model (shown in Figure. 4) is by an order of magnitude greater.

Fig. 7 Simultaneous fitting of the transmittance (a) and reflectance (b) change using the nonuniform model. The black dots show the experimental results of the pump–probe experiment at the 5 ps delay time; the red-line denotes the fitting of the nonuniform model.

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From the model fitting, we retrieved the real and imaginary parts, $Δ ε_{fcr}^{r} (z, λ)$ and $Δ ε_{fcr}^{i} (z, λ)$ , of the dielectric function change 5 ps after the excitation by the pump. These functions are shown in Figure 8. It can be seen that $Δ ε_{fcr}^{r}$ and $Δ ε_{fcr}^{i}$ are only weakly dependent on the probe wavelength, as expected for the free carrier response measured away from the resonance. However, these functions depend strongly on the position, z: $Δ ε_{fcr}^{r} (z)$ is changing from around −0.035 to −0.012, while $Δ ε_{fcr}^{i} (z)$ is shifting from around 0.022 to 0.003 as a function of the depth between 0μm and 13μm. The observed change corresponds to the excitation profile decaying as a function of the distance from the front surface. Moreover, by comparing Δε_fcr with the dielectric function of nano c-Si without pump excitation ε_c–Si in Fig. 3, it can be seen that the fractional change of the real part, $Δ ε_{fcr}^{r}$ , is quite small in comparison to that of the imaginary part, $Δ ε_{fcr}^{i}$ . Such a relation is characteristic for the free carriers response when the plasma frequency and scattering rate are of the same order [36].

Fig. 8 Wavelength-dependent lateral distribution of the real (a) and imaginary (b) parts of the dielectric function change induced by the pump excitation, retrieved from the optical model based on the WKB approach.

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We note here that the profile of the excitation strength (the dielectric function change) is in general different from that expected for a semi-infinite solid because of the pump pulse absorbance enhancement resulting from bouncing inside the membrane between the rear and front surfaces. It can be estimated that the pump pulse travels 2 −−3 round trips before its intensity decays by a factor of 1/e.

4. Conclusion

In conclusion, we investigated and developed an analytical calculation method to estimate the spatial distribution along the membrane depth of the dielectric function of porous silicon membranes excited optically by a femtosecond laser pulse. We found that a model with a uniform excitation is inadequate as it does not provide a solution to reconstruct the observed experimental results and neglects the spatial excitation profile of the free charge carriers. To overcome this problem we developed an analytical method based on the WKB approximation. Using this method, we obtained the spatial distribution of the real and imaginary parts of the dielectric function bidirectionally coupled by the Kramers–Kronig relation. The results confirm that the main contribution to the change of the dielectric function induced by the optical pump can be attributed to the excited free carrier response. Thus, the obtained spatial distribution conveys the distribution of the excited carriers as a function of the membrane depth. Moreover, it seems possible to manipulate and to engineer the dielectric function by tuning the pump pulse intensity.

The modelling results provided an insight that the fractional change of the imaginary part, associated with the free carrier absorption, is significantly stronger than that of the real one. It also suggests that the excitation profile of the membrane differs from that of its semi-infinite solid counterpart because for the former the round trips and constructive interference of the pump pulse enhance the absorbance.

We believe that the method of reconstructing the spatial dielectric function distribution is useful not only for optically pumped materials but can be applied to biosensing where the PS refractive index is sensitive to the binding of biological and chemical molecules for which adsorption and distribution inside a thin film are strongly non-uniform, solar cells and detectors with a nonuniform structure and morphology, and a variety of electro-optical devices with induced or permanent nonuniform dielectric function modulation. In general, this model is applicable for the samples sufficiently transparent and reflective to measure observable values of the transmittance and reflectance, respectively.

Acknowledgments

The Coherent laser system used in this research was obtained through the Birmingham Science City project: Creating and Characterising Next Generation Advanced Materials, supported by Advantage West Midlands (AWM) and funded in part by the European Regional Development Fund (ERDF). We thank T. Roger, J. Barreto and D. Chekulaev for help with experimental setup. We thank EPSRC and DSTL for the financial support.

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Determination of excitation profile and dielectric function spatial nonuniformity in porous silicon by using WKB approach

Abstract

1. Introduction