1. Introduction

Owing to high sensitivity, fast response, no labeling, and no invasion, sensors utilizing surface plasmon resonance (SPR) have been widely employed [1,2]. SPR is a coupled and localized electromagnetic surface wave propagating along the medium interface and decaying exponentially [3]. When SPR is excited, the photon energy of incidence is strongly absorbed and a sharp dip shows up in the spectral–directional reflectance spectrum. The dip can shift almost instantly with little variation in the optical constants of media, so high sensitivity and fast responses for biomedical and chemical applications are guaranteed [4]. Current configurations of SPR-based biomedical and chemical sensors vary a lot, while the core element is a waveguide, a prism, or gratings to excite SPR [5]. Since appropriate materials for core elements are limited, most known SPR-based sensors work in the visible [6,7] or near-infrared (near-IR) [8,9] region.

However, the mid-IR region is a very important spectral region in which to catch different target materials for biomedical and chemical applications [10–13]. The list of applicable target materials will certainly expand greatly if mid-IR SPR-based sensors are developed. Smooth contact interfaces of these factors are specifically desired because the roughness easily prohibits the liquid flow or activities of living cells. Therefore, this work proposes a type of mid-IR SPR-based sensor, which can excite SPR at specified wavelengths and minimize side effects resulting from an uneven surface. Of course, the sensors should function with a simple optical setup and withhold all advantages of existing ones.

The keys to sensor development are tailored optical constants of highly doped silicon and a well-controlled doping profile. When the doping concentration is above 1×10¹⁹ cm^-3, SPR can be excited with a periodic profile in the IR region regardless of dopant types [14,15]. Such a high doping concentration can be realized by ion bombardment of lightly doped silicon followed by annealing for dopant activation [15]. SPR wavelength is then finely tuned with optical constants and wavevector components, which differ in doping concentration and doping profile [16–18]. If the doping profile is embedded in the intrinsic silicon and the mirror-like surface is kept, the drawbacks of uneven surfaces will diminish. In this work, an efficient algorithm, rigorous coupled-wave analysis (RCWA) [19], will be employed for numerically illustrating the feasibility and performance of developed sensors. The working spectral region of sensors will focus on the mid-IR region, while the same idea can be extended to a broader region.

2. Excitation of a mid-IR surface plasmon polariton

SPR excitation depends on the polarization of incidence, wavevector components, and optical constants of associated media [20]. For transverse magnetic (TM) wave incidence, the requirement for SPR excitation between two semi-infinite media is [21]

where k is the magnitude of wavevector k, which can be decomposed into a parallel component (k _∥) and a perpendicular component (k _⊥) with respect to the interface. ε = (n + iκ)² is the dielectric function of a medium, where optical constants n and κ are the refractive index and extinction coefficient, respectively. ω represents the angular frequency of incidence, and c is the light speed in vacuum. Note that the dielectric function and optical constants are functions of wavelength λ. Accordingly, optical constants of sensor materials and the working medium should be studied to search general guidelines for developing sensors.

In this work, the spectral range of interest is mainly the mid-IR region (2 μm ≤ λ ≤ 20 μm) and the working medium for developed sensors is either free space or water. Optical constants of free space are assumed to have the same constants as those in vacuum (n = 1 and κ = 0). On the other hand, tabulated optical constants of water in [22] and those of intrinsic silicon in [23] are adopted with appropriate interpolation. In contrast to those of intrinsic silicon, optical constants of highly doped silicon change a lot with the dopant type, wavelength, and doping concentration. The Drude model employed here for optical constants of highly doped silicon has more accurate identification on carrier mobility and ionization [19,22]. Only the phosphorous-doped silicon at doping concentration (N _P) at 1×10²⁰, 5×10²⁰, and 1×10²¹ cm^-3 are discussed to reduce the redundancy.

 

Fig. 1. Optical constants of water, intrinsic silicon, and highly doped silicon. (a) Refractive index, n; (b) extinction coefficient, κ.

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Figure 1 shows plots of optical constants of intrinsic silicon, highly doped silicon, and water in the spectral region of interest. The refractive indices of intrinsic silicon and water almost remain constant, but the refractive indices of highly doped silicon show significant variation. At N _P = 1 × 10²⁰ cm^-3, the minimum and maximum refractive indices are n = 1 at λ = 7 μm and n = 4 at λ = 20 μm, respectively. As the doping concentration increases, the minimum shifts to shorter wavelengths and the maximum at λ = 20 μm enlarges monotonically. In Fig. 1(b), the extinction coefficients of intrinsic silicon and water are less than 10^-3 and 10^-1, respectively. Note that absorption in free space is not considered in this work because the extinction coefficient is set to zero. In contrast, the extinction coefficient of highly doped silicon augments monotonically with the wavelength and even surpasses the refractive index. Plots in Fig. 1 point out several advantages of using doped silicon for the development of sensors. For one, the real part of dielectric function n ² - κ² becomes negative and can support SPR excitation in free space or water, in which real parts of dielectric function are positive. Furthermore, the transition wavelength associated with the real part of dielectric function can be manipulated to offer flexibility in tuning the SPR wavelength. Besides, the low-extinction coefficients at short wavelengths also favor SPR excitation with negligible loss.

 

Fig. 2. Dispersion curves between semi-infinite, highly doped silicon and a working media. (a) Free space; (b) water.

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Figures 2(a) and 2(b) plot dispersion curves of semi-infinite, highly doped silicon in free space and in water, respectively. Although dispersion curves are almost identical by setting either k_∥ or ω to be a real number [20], a real co and the real part of k// are employed here for plotting curves. The units of ω and k_∥ are converted to cm^-1 by dividing 2πc and 2π, respectively. In Fig. 2(a), the y axis and the long dashed curve represent light lines at different angles of incidence θ = 0° and θ = 45°, respectively. The other three curves represent dispersion curves for silicon at selected doping levels. The three curves partly overlap at low frequency and show similar characteristics at the same frequency. For example, flips of curves at ω ≈ 1600 and 3300 cm^-1 in Fig. 2(b) come from the extinction coefficient peaks of water. In principle, SPR is excited when the light line crosses the dispersion curve so that the discrepancy among each dispersion curve provides multiple choices for the same target material. In this work, mid-IR SPR-based sensors are developed with a periodic doping profile, which functions like gratings, generating diffraction orders with a sufficiently large wavevector.

3. Optical setup and sensor design guidelines

Figure 3 shows the configuration of the optical setup working with the proposed sensors. Other elements of the system include a mid-IR light source, a linear polarizer, and a detector. The plane of incidence is the x-z plane, while E and H are the electric and magnetic fields, respectively. The TM wave from the light source and polarizer is incident on the sensor surface, and the reflectance is collected by the detector. The developed sensors with the optical setup can work in three ways like the existing ones [6]. One is used to scan a wide spectrum routinely and exhibits the variation of the reflectance dip wavelength. The second is similar to the first one, except that the wavelength is fixed and the objective of scanning becomes the angle of incidence. The third is used to monitor the reflectance fluctuation in real time at a fixed angle of incidence and wavelength. Though the incidence and reflectance interfere with each other at normal incidence, utilization of beam splitters and mirrors can be a good solution. The magnified cross-sectional view shows a periodic doping profile above a highly doped silicon film. The doping profile repeats itself in the x direction and extends in the y direction to cover the incidence beam spot. In the numerical model, the profile is considered a one-dimensional periodic binary grating.

The sensor can be fabricated with a commercially available wafer because it is composed only of intrinsic and highly doped silicon. The doping profile is a synthesized pattern by doping from both top and bottom sides of a polished thin film. At the top side, a periodic sacrificial layer is patterned to shade the film during doping. Then, a periodic doping profile embedded in the film is formed after removing the sacrificial layer. The bottom side is doped without any sacrificial layer coverage and the doped region merges with that from the top side after annealing. Since the diffusion length of dopants is much longer than the penetration depth of incidence, the highly doped silicon at the bottom side of the sensors is thick enough to be assumed as semi-infinity. Note that the penetration depth (λ/4πκ) of highly doped silicon is usually less than 1 μm in the mid-IR range. While challenges may still exist in real fabrication, an ideal doping concentration and doping profile is assumed in this work. The periodic profile is specified with three dimensions: the period (Λ), lateral filling ratio (f), and depth of intrinsic silicon (d _g).

 

Fig. 3. Configuration of the optical setup with a developed mid-IR SPR-based sensor in a working medium. The magnified cross-sectional view of the sensor is also included.

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The SPR wavelength of developed sensors should be adjustable to fit target materials, so general guidance is imperative. In contrast to prisms limited by available materials, SPR excitation by periodic profiles can tailor the wavelength with great flexibility by flipping dispersion curves to intersect light lines [24]. Figures 4(a) and 4(b) show dispersion curves for semi-infinite silicon at N _P = 1×10²¹ cm^-3 in free space and in water, respectively. These curves are good approximations to those of sensors with a periodic doping profile because the wavelength is much longer than the size of the features. Since the difference among the k_∥ of each diffraction order is a multiple of 2π/Λ, the dispersion curves are bent into a region 0 ≤ k_∥ ≤ π/Λ due to the periodicity and symmetry [21]. The bending to the reduced zone scheme for the even function f(k_∥) is given by [20]

where j is the reflective diffraction order. σ = 1 if j is a non-negative integer; otherwise, σ = -1. That is, branches from the bottom represent diffraction orders in the sequence of 0, -1, +1, -2, +2, and so on. Due to the truncation of the y axis for clarity, the first branch of the dispersion curve for Λ = 10 μm is not shown in Fig. 4(a). The maximum k_∥ of each reduced zone is determined by the doping profile period with k_∥ = π/Λ, which is marked with a vertical dotted line. In Fig. 4(a), the solid curve with solid square marks and the dashed-dotted curve with circle marks are for Λ = 4 μm and Λ = 10 μm, respectively. The first vertical line is also used in Fig. 4(b) to symbolize the same doping period, Λ = 4 μm. On the other hand, the dashed-dotted curve with circle marks and the dashed curve with square marks in Fig. 4(b) are for Λ = 2 μm and Λ = 3 μm, respectively. Most light lines should be flipped like dispersion curves, so that the dashed curve sections in two reduced zones of Fig. 4(a) come from the light line of θ = 45° for Λ = 4 μm and Λ = 10 μm.

 

Fig. 4. Flipped dispersion curves in reduced zones for highly doped silicon (N _P = 1×10²¹ cm^-3) sensors with a different doping profile period (Λ). (a) Sensors working in free space with Λ = 4 and 10 μm; (b) sensors working in water with Λ = 2, 3, and 4 μm.

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Intersection points of the dispersion curve and light line in the reduced zone specifies the SPR frequency/wavelength and corresponding angle of incidence. For example, the lowest frequency of intersection points between the y axis and the dispersion curves in Fig. 4(a) are 1000 cm^-1 and 2500 cm^-1 for Λ = 10 μm and Λ = 4 μm, respectively. Consequently, the reflectance dip is expected to be Λ = 10 and λ = 4 μm, with the j = ±1 order diffractions exciting SPR at θ = 0°. Here the SPR wavelength coincides with the profile period because the dispersion curves are not seriously perturbed by optical constants. At oblique incidence, such as θ = 45°, the lowest frequency of intersection points are about 580 and 1400 cm^-1 for Λ = 10 and 4 μm, respectively. The expected SPR wavelengths are therefore λ = 17.2 μm and λ = 7.1 μm. In general, SPR can also be excited at the frequency corresponding to other intersections, while these reflectance dips are insignificant due to weak diffractions of higher orders. Hence, only the lowest intersection frequency is considered for SPR-based sensors.

Figure 4(b) gives a guideline for developed sensors working in water with different doping profile periods. The period Λ = 4 μm is shown here as a reference for that in Fig. 4(a), while other periods in Fig. 4(b) are Λ = 2 μm and Λ = 3 μm. The lowest frequency of intersection points between the y axis and the dispersion curves are about 1900, 2500, and 3850 cm^-1 for Λ = 4, 3, and 2 μm, respectively. The reflectance dip is accordingly expected to be λ = 5.3 μm, λ = 4 μm, and λ = 2.6 μm at normal incidence. Note that λ here refers to the wavelength in vacuum and the real wavelength in water should be reduced by a factor of (ε_w)^-1/2, where ε_w is the dielectric function of water. Accordingly, the SPR wavelength is not the same as the doping profile period. The high extinction coefficient of water brings large loss to SPR such that the reflectance dip may blur or even disappear. For example, having sensors working around λ = 6 μm is very challenging due to the strong absorption at 1595 cm^-1 for H₂O vibration modes [24]. Moreover, the oscillation of the extinction coefficient forms bumpy dispersion curves, which may lead to a discrepancy between the predicted and real wavelength of SPR excitation.

4. Applicable scope

Figure 5 shows the reflectance spectra of developed sensors working in free space at both normal and oblique incidence. The spectra are obtained from programmed codes based on RCWA, which can efficiently acquire radiative properties of micro/nanoscale periodic structures. The computer codes programmed in the author’s group have been validated numerically [25] and experimentally [26]. For clarity, the reflectance spectrum is not fully exhibited, and each line type correlates to one doping profile period. The depths of intrinsic silicon d _g are optimized to show sharp reflectance dips, while the lateral filling ratio f = 0.5 is fixed. Both Λ and d _g are specified in the legend, while the profile period is set to Λ = 4, 6, 8, 10, 12, 14, or 16 μm. Not all reflectance spectra of sensors with selected profile periods are plotted because SPR is not always excited, especially with large loss.

Figures 5(a) and 5(b) show reflectance dips for sensors made of silicon at N _P = 1×10²¹ cm^-3, and the angle of incidence is θ = 0° and θ = 45°, respectively. The periods of the doping profile is Λ = 4, 6, 8, or 10 μm. In Fig. 5(a), the wavelength of the reflectance dips exactly match those predicted from the dispersion curves. The relative reflectance difference is higher than 0.25 within ±2 μm the spectral region for all dips shown. The optimized d _g enlarges with the increment of the doping period and dip wavelength. When Λ ≥ 12 μm, the reflectance dip becomes trivial due to the loss and thus is not shown. In Fig. 5(b), the optimized d _g is shallower than that in Fig. 5(a) and the reflectance dip becomes wide. High sensitivity of those sensors is still promising because the reflectance variation of a dip is larger than 0.2. The SPR wavelengths at oblique incidence also follow the guideline based on flipped dispersion curves. For instance, the wavelength of reflectance dips for Λ = 4 μm and Λ = 10 μm are λ = 7.1 μm and λ = 17.2 μm, respectively.

Sensors in Figs. 5(c) and 5(d) are similar to those in Fig. 5(a) ad 5(b), except the doping concentration is lower (N _P = 5×10²⁰ cm^-3) and optimized d _g differs. Since dispersion curves for silicon at N _P = 1×10²¹ cm^-3 and N _P = 5×10²⁰ cm^-3 in free space are quite similar, the guideline given previously from silicon at N _P = 1×10²¹ cm^-3 should still be valid. However, the smaller extinction coefficient for N _P = 5×10²⁰ cm^-3 means relatively low loss and facilitates SPR at long wavelengths. Therefore, the reflectance dip at λ = 12 μm is realized with sensors for Λ = 12 μm and NP = 5×10²⁰ cm^-3. In Fig. 5(c), five sharp dips (at λ = 4, 6, 8, 10, and 12 μm) are drawn, and their reflectance variations are larger than 0.25. In contrast, only four dips are able to be plotted in Fig. 5(d) because the other dip is outside the spectral range. The optimized d _g still increases with the SPR wavelength. Similarly, the optimized d _g is larger at normal incidence than that at oblique incidence for sensors of the same Λ.

 

Fig. 5. Reflectance dips for sensors working in free space at different doping concentrations (N _P) and angle of incidence (θ). (a) N _P = l×l0²¹ cm^-3 and θ = 0°; (b) N _P = l×l0²¹ cm^-3 and θ = 45°; (c) N _p = 5×l0²⁰ cm^-3 and θ = 0°; (d) N _P = 5×l0²⁰ cm^-3 and θ = 45°; (e) N _P = l×l0²⁰ cm^-3 and θ = 0°; (f) N _P = 1×l0²⁰ cm^-3 and θ = 45°.

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The wavelengths of reflectance dips shown in Figs. 5(e) and 5(f) also match those from dispersion curves of the silicon with a higher doping concentration. The optimized d _g remains increased with Λ, but the optimized d _g of sensors with Λ = 8 μm at normal incidence is 0.4 μm, which is much smaller than d _g = 1.3 μm at oblique incidence. Besides, the optimized d _g of sensors at N _p = 1×10²¹ cm^-3 is less than that of sensors at a higher doping concentration at normal incidence. Since the largest optimized depth is d _g = 1.7 μm, which is about one order of magnitude smaller than λ, dispersion curves between two semi-infinite media are still applicable. In both figures, the shortest wavelength of reflectance dips is λ = 6 μm because the real part of the silicon dielectric function is positive and unable to excite SPR at shorter wavelengths. In Fig. 5(e), the dip is more significant than those in Figs. 5(a) and 5(c) with reflectance variation larger than 0.4. In contrast, the reflectance variation in Fig. 5(f) is similar to that in Figs. 5(b) and 5(d) and less than 0.3. Three dips are shown in Fig. 5(f) and their wavelengths are identical to those with higher doping level. The dip at λ = 17.2 μm, on the contrary, does not exist due to the large absorption. According to reflectance dips in Fig. 6, none of selected doping concentration can excite SPR in the whole spectral range of interest.

Figure 6 shows the reflectance dips of sensors made of silicon at N _P = 1×10²¹ cm^-3. Since the working medium is water, absorption is strong and the applicable spectral region for sensors is limited to λ ≤ 6 μm. In Fig. 6(a), SPR can be excited at λ ≈ 2.5, 4.1, and 5.2 μm by sensors with Λ = 2, 3, and 4 μm, respectively. These wavelengths are close to those predicted from the dispersion curves. As the period enlarges, the absorption gets strong and the reflectance dip becomes insignificant. Though the optimized d _g increases with the SPR wavelength as those of sensors working in free space, the reflectance variations of dips here are much less. In Fig. 6(b), two reflectance dips of sensors with Λ = 10 and Λ = 12 μm are both plotted at θ = 45°. Clearly, mid-IR SPR-based sensors with large Λ can still work in water at oblique incidence.

 

Fig. 6. Reflectance dips for sensors working in water at doping concentration N _P = 1×10²¹ cm^-3 and different angle of incidence (θ). (a) θ = 0°; (b) θ = 45°.

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5. Performance demonstration

The performance of developed sensors is numerically examined and demonstrated in three ways. One way is to plot the complete reflectance spectra for a developed sensor as well as for sensors with little modifications in intrinsic silicon depth and metallic filling ratio. In that case, the physical mechanism for the reflectance dip can be confirmed because dips by other physical mechanisms are usually sensitive to the geometry variation [19]. Figure 7(a) shows the reflectance spectra associated with a developed SPR-based sensor, which works in water at normal incidence with Λ = 3 μm and N _P = 5×10²⁰ cm^-3. The intrinsic silicon depth of sensors is d _g = 0.3, 0.4, or 0.5 μm, while the filling ratio is f = 0.2, 0.5, or 0.8. The complete reflectance spectra of 2 μm ≤ λ ≤ 12 μm are shown and the target SPR wavelength is set to λ = 4 μm, matching with the prediction from the dispersion curves. It is obvious that all spectra show a reflectance dip at λ = 4 μm, regardless of d _g and f. In contrast, other dips in the reflectance spectrum shift their wavelengths and magnitudes significantly because they are irrelevant to SPR. In addition to confirming the reflectance dip caused by SPR, Fig. 7(a) also exhibits two superiorities of developed sensors. First, the reflectance dip at λ = 4 μm is not affected by other dips, so signals from developed sensors can precisely reflect SPR excitation. Second, the performance of developed sensors allows flexibility in doping profiles, which is very critical to fabrication and commercialization. The above discussion is summarized from numerous sensors with geometric parameters that are not shown here.

 

Fig. 7. (a) Complete spectral reflectance spectra for associated sensors of the same doping concentration (N _P = 5×10²⁰ cm^-3) and period (Λ = 3 μm), but the filling ratio (f = 0.2, 0.5, and 0.8) and the intrinsic silicon thickness (d _g = 0.3, 0.4, and 0.5 μm) are different. The working medium is water and the SPR is excited at λ = 4 μm for all sensors. (b) Developed sensors successfully catch a target material, an aqueous ammonium sulfate binary mixture (20% weight fraction (NH₄)₂SO₄) at 298 K.

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Another sensor performance demonstration involves catching real chemicals or biomedical samples. The 20% weight fraction aqueous ammonium sulfate binary mixture (NH₄)₂SO₄ is selected as an example of targets owing to available optical constants [27]. Figure 7(b) shows the reflectance spectra of two developed sensors in the mixture and water. The filling ratio and doping concentration of both sensors are f = 0.5 and N _P = 1×10²⁰ cm^-3, respectively. On the other hand, d _g, Λ, and θ of sensors differ largely, so the aiming SPR wavelength is not the same. When the working medium changes from water to the binary mixture, the SPR wavelength clearly shifts. The SPR wavelength associated with the binary mixture gets longer than that associated with water for both sensors. This case, indeed, gives strong support to the claimed capabilities of developed sensors operating in the spectral region of interest.

A third way to evaluate the performance of developed sensors is to inspect the instrumental contribution to sensor sensitivity, S_RI1 = δλ_SPR/δn, which is the ratio of dip wavelength variation with respect to the change in refractive index [6]. Figure 8 shows the reflectance spectra of three representative developed sensors with different doping profiles and doping concentrations. The direction of incidence, SPR wavelength, and working media are not the same, so the sensor assessments drawn here are convincing. Optical constants of working media vary in the whole spectral range, and the corresponding reflectance spectra are plotted. For the free space as the reference working medium, the dielectric functions in Figs. 8(a) and 8(b) increase in a step size of 2%. On the other hand, the dielectric functions of the medium in Fig. 8(c) may either increase or decrease from that of water, which is the reference working medium here. When the dielectric function increases, the SPR wavelength (λ_SPR) corresponding to the dip also increases. Since the magnitude of the wavelength shift is not determined by single factor, the sensitivity of each sensor differs a lot. S_RI1 of three sensors in Figs. 8(a), 8(b), and 8(c) are about 4100, 17200, and 900 nm/RIU, respectively. These values successfully exhibited the superiority of proposed sensors because S_RI1 of most grating coupler-based sensors is less than 750 in the visible and near-IR range [5]. Note that ε = (n + iκ)², and three sensors shown in Fig. 8 are not optimized. As a result, practitioners must be cautious in selecting developed sensors among multiple choices with appropriate doping profiles, doping concentrations, and the directions of incidence.

 

Fig. 8. Reflectance spectra for selected sensors working in a medium with modified dielectric function based on a reference medium. The dielectric function ratio is 1.01, 1.03, 1.05, or 1.07 for free space, while it is 0.97, 1.01, 1.03, or 1.05 for water. The characteristics of sensors are (a) N _P = 1×10²¹ cm^-3, Λ = 4 μm, d _g = 0.4 μm, f = 0.5, θ = 0°; (b) N _P = 5×10²⁰ cm^-3, Λ = 10 μm, d _g = 0.5 μm, f = 0.5, θ = 45°; (c)N _P = 1×10²⁰ cm^-3, Λ = 2 μm, d _g = 0.2 μm, f = 0.5, θ = 0°.

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6. Conclusion

This work has successfully developed a new type of biomedical and chemical mid-IR SPR-based sensor working in either free space or water. The sensor is a periodic doping profile embedded in a silicon film, and the optical constants are tailored with the doping concentration. Dispersion curves between doped silicon and working media provide sufficiently accurate guidance in tuning the SPR wavelength to monitor the target material with a reflectance dip. The depth of intrinsic silicon is optimized to obtain a significant reflectance dip. Based on the proposed optical setup and design guideline, numerical results clearly demonstrate the wide working spectral range for developed sensors. Furthermore, multiple sensor choices are always available for any target wavelength by adjusting doping characteristics. Other advantages of developed sensors include a smooth contact interface, tolerance in doping profile, and high sensitivity.