The Optimization of Metal Nitride Coupled Plasmon Waveguide Resonance Sensors Using a Genetic Algorithm for Sensing the Thickness and Refractive Index of Diamond-like Carbon Thin Films

Abstract

We theoretically designed the Kretschmann configuration coupled plasmon-waveguide resonance (CPWR) sensors, composed of thin films of metal nitrides. The thicknesses of the layers of the CPWR sensors were optimized using a genetic algorithm. The optimized CPWR sensors were applied to simultaneously measure the thickness and refractive index (RI) of diamond-like carbon (DLC) films. The field profiles and the sensitivity of the CPWR sensors in response to thin DLC films were studied using the finite-different time-domain technique and the transfer matrix method. The genetic algorithm method predicted that the two-mode CPWR sensors could simultaneously analyze the thickness and RI of the DLC films as thin as 1.0 nm at a wavelength of 1550 nm. The simulations showed that the angular sensitivity toward the refractive index changes of the DLC films of the optimized CPWR sensors was comparable to that of traditional CPWR sensors.

1. Introduction

Surface plasmons are surface electromagnetic waves that propagate along the metal-dielectric interface. The surface plasmon resonance (SPR) occurs when the wave vector of the surface plasmon is matched with the wave vector of the incident light on a prism in the Kretschmann configuration [1]. Surface plasmons are responsible for several fascinating phenomena that have been employed in a variety of applications [2,3]. Recently, SPR techniques have been used for applications such as the perfect absorbers [4,5] and thickness sensors of thin film overlayers on the active surface plasmon layer [6,7,8]. Typical SPR methods for assessing the thickness of thin film include measuring the reflected intensity as a function of wavelength or angle and finding the positions of the reflectivity minimum, which relate to both the refractive index (RI) and the thickness of the thin film to be measured. The traditional SPR sensor consists of a prism with a noble metallic film coating deposited on top, which is adjacent to a dielectric medium, such as air or water. The traditional coupled plasmon-waveguide resonance (CPWR) architecture consists of a waveguide layer on top of the noble metallic film of the traditional SPR structures [9]. Conventional CPWR sensors have narrow full width at haft maximum and sharp waveguide resonance, and they can be excited by both transverse magnetic (TM) and transverse electric (TE) polarized light [10]. Due to the greater containment of surface plasmons in the waveguides, conventional CPWR sensors have potential advantages over traditional SPR sensors [11]. The CPWR sensors have been used in a range of applications, including chemical and biological sensing [12,13,14], as well as thickness and refractive index (RI) change detection [15,16,17,18]. The traditional CPWR and SPR devices create broad resonances at near-infrared (NIR) wavelengths due to the high ohmic loss of the noble metal films, limiting their usage at these wavelengths [19]. Moreover, the noble metal films have a low damage threshold and lack of compatibility with standard complementary metal oxide semiconductor (CMOS) technology [20]. Thus, the alternative plasmonic materials have been developed to overcome these drawbacks [21,22,23,24,25].

Thin films of titanium nitride (TiN), the alternative plasmonic material, have attracted significant interest for plasmonic applications in recent years due to their plasmonic properties, which are comparable to thin films of gold in the wavelength range of more than 500 nm [26,27,28], and their compatibility with the CMOS fabrication technique [29]. Moreover, the TiN films have tunable optical properties and can resist higher temperatures than gold films [30]. Although thin films of TiN have promised plasmonic performance, there have been few studies on their application in SPR sensors. Some of these studies included numerical investigation of the sensitivity of the TiN/sapphire SPR sensors [31], TiN-coated photonic crystal fiber sensor [32] and experimental demonstration of the responses of sapphire/TiN sensors [33], and fluorescence sensing [34].

To obtain the best performance, the sensitivity and the RI resolution of the SPR sensors were optimized using many optimization techniques including the particle swarm optimization [35,36] and genetic algorithms (GA) [37,38,39]. Although the GA was utilized to optimize the wavelength of the operation and layer thicknesses of the CPWR sensors for biochemical sensing, the waveguide supported only a single resonance for TE and TM modes. The layer structures of the CPWR sensors that can generate multi-resonance dips were not optimized for optimal response.

The objective of this paper was to design the CPWR sensors, which consisted of thin films of TiN and aluminum nitride (AlN), to measure simultaneously the thickness and RI of the diamond-like carbon (DLC) thin films in the NIR spectral range. The DLC thin films were chosen as an example for simultaneously determining thickness and RI because they are extensively used in many industrial applications [40] and they can be grown on the AlN layer without causing any structural changes [41]. To achieve the aim, the CPWR sensors composed of the TiN and AlN layers were designed and optimized to produce either two or three TM resonance minima at a wavelength of 1550 nm using the GA method. The plasmonic response and electrical field profiles of the CPWR structures were investigated using the transfer matrix method (TMM) and a finite-different time-domain (FDTD) technique for the different thicknesses of the DLC thin films. The FDTD technique was utilized to simulate the optical effects of a variety of nanophotonic devices. It has been shown that the FDTD technique allows efficient and precise simulations of spectral responses in plasmonic systems [42,43].

2. Materials and Methods

The design of the CPWR sensors is discussed first, followed by the theoretical models used to analyze them.

2.1. Selection of Materials for CPWR Sensor

Our design was based on the optical loss resulting from the intraband transitions in the NIR spectral range and the compatibility with CMOS fabrication processes of TiN films. To obtain the best plasmonic performance and sensor efficiency, TiN films require suitable substrates and must be integrated with III-V nitrides, such as AlN and GaN, with smooth interfaces and surfaces. Previous studies have shown that TiN thin films can be fabricated on c-sapphire, magnesium oxide (MgO), and silicon substrates. Previous research has also shown that TiN films grown on sapphire substrates had better plasmonic performance than TiN films grown on silica glass or silicon substrates [44,45]. In this work, we examined the CPWR sensors on sapphire, magnesium oxide, and silicon prisms. We focused on the lattice match with TiN while choosing the III-V nitrides as a waveguide. The AlN film, having a lattice constant of 4.08 Å, was chosen because of the low lattice mismatch with TiN, having a lattice constant of 4.24 Å [46]. Moreover, experiments revealed the good adhesion property of the TiN/AlN bilayers [47,48].

Figure 1 shows a schematic representation of the CPWR structures to be studied without and with a thin DLC layer. The CPWR sensor consisted of a prism that was deposited with TiN, and an AlN layer was coated on top of the TiN layer (Figure 1a). A thin DLC film to be evaluated for RI and thickness was deposited on the AlN layer (Figure 1b). The SPR effect is observed when the wave vector of the surface plasmon (k_sp) is matched with the wave vector of the incident light (k_x,TM) on the prism. The CPWR sensors were in the ambient air.

2.2. Theoretical Modeling

2.2.1. Transfer Matrix Method (TMM)

The Fresnel’s equations and the TMM were used to describe the reflectivity as a function of the angle of incidence of the CPWR structures. The theoretical reflectivity spectra of the CPWR structures were calculated using the MATLAB program. The CPWR system can be modeled as a four-phase model: prism/TiN/AlN/air. Although the CPWR systems can support both TE and TM modes, the SPP can only be created by the TM mode. The details of the TMM method are given in the literature [9,49]. Below, we briefly describe the TMM approach for the CPWR system.

Consider the stack of layers in z-axis direction in the CPWR system, as shown in Figure 1b. The prism and the surrounding medium (air) are numbered 1 and 4, respectively. The films are numbered from 2 to 3, while the interfaces are numbered from 1 to 3. The first interface between the prism and TiN was defined as $z=z_1=0$, and the plane of incident light as the xz plane. The amplitudes of the electric and magnetic fields, E and H, in the first layer, resulting from the Jth layer, are calculated by multiplying each of the matrices for separate layers as follows [49]:

$$\left[\begin{array}{l}{H}_1\\ E_1\end{array}\right]={\displaystyle {\prod }_{j=1}^{J-1}{M}_j}·\left[\begin{array}{l}{H}_j\\ E_j\end{array}\right]$$

(1)

The matrix M_j is given by

$$M_j=\left[\begin{array}{cc}\cos {\varphi }_j\hfill & -i\sin {\varphi }_j/q_j\\ -i{q}_j\sin {\varphi }_j\hfill & \cos {\varphi }_j\end{array}\right]$$

(2)

where ${\varphi }_j$ is the phase factor in the Jth layer, which can be calculated by

$${\varphi }_j=\left(2\pi /\lambda \right)\left(n_j-i{k}_j\right)d_j\cos {\theta }_j$$

(3)

where $\lambda$ is the wavelength of the incident light, $n_j$ and $k_j$ are the refractive index and extinction coefficient of the Jth layer,$d_j=z_j-z_{j-1}$ is the thickness for the Jth layer, and ${\theta }_j$ is the incident angle of the Jth layer, respectively, and

$$q_j={\left({\mu }_j/{\epsilon }_j\right)}^{1/2}\cos {\theta }_j$$

(4)

The reflectance of the CPWR system is calculated by

$$R={\left|\frac{\left(m_{11}+m_{12}{q}_N\right)q_1-\left(m_{21}+m_{22}{q}_N\right)}{\left(m_{11}+m_{12}{q}_N\right)q_1+\left(m_{21}+m_{22}{q}_N\right)}\right|}^2$$

(5)

where $q_1$ and $q_N$ are calculated from Equation (4). For SPR sensors without a DLC coating layer on the TiN layer, $q_N=q_3$; for CPWR sensors without a DLC deposition layer on the AlN layer, $q_N=q_4$; and for CPWR sensors with a DLC deposition layer on the AlN layer, $q_N=q_5$.

2.2.2. Drude–Lorentz Model

The dielectric permittivity of the TiN films in the wavelength range of 500–2000 nm were obtained by fitting the experimentally measured spectroscopic ellipsometer curve to the Drude–Lorentz model [29]:

$$\epsilon \left(\omega \right)={\epsilon }_{\infty }-\frac{{\omega }_p^2}{{\omega }^2+i{\Gamma }_D\omega }+{\displaystyle {\sum }_{j=1}^3\frac{{\omega }_{L,j}^2}{{\omega }_{0,j}^2-{\omega }^2-i{\gamma }_j\omega }}$$

(6)

where ${\epsilon }_{\infty }$ is the high-frequency dielectric constant, ${\omega }_p$ and ${\Gamma }_D$ are the plasma frequency and Drude damping constant, respectively. ${\omega }_{L,j}^2=f_j{\omega }_{0,j}{\gamma }_j$, where $f_j,{\omega }_{0,j}$ and ${\gamma }_j$ describe the Lorentz oscillator strength, resonant energy, and damping factor, respectively.

The permittivity of TiN films is derived from fitting one Drude and one Lorentz term using a nonlinear least-squares method with ${\epsilon }_{\infty }=4.7988$, ${\omega }_p=7.7557$, ${\Gamma }_D=0.15586$, ${\omega }_L=5.3154$, ${\omega }_0=3.9169$, and $\gamma =1.7222$.

2.3. The Finite-Difference Time-Domain (FDTD) Method

Numerical simulations were conducted using the two-dimensional FDTD method with the program OptiFDTD [50]. In the FDTD simulations, the CPWR structures, covering a region of 6 μm × 6 μm, were calculated under the incident plane wave with perfectly matched layers and a two-dimensional cell size of 3 nm. For all simulations, the wavelength was fixed at 1550 nm. The model parameters for the CPWR configurations are summarized in

Table 1. Summary of parameters used for simulation the CPWR sensors at 1550 nm.

Layer	Material	Refractive Index
Prism	Sapphire	1.746 [51]
	Silicon	3.48 [28]
	Magnesium oxide	1.715 [52]
Plasmon active	Titanium nitride	0.9857 + 9.1899i [29]
Waveguide	Aluminum nitride	2.0293 + 0.0001i [53]
Sensing	Dimond-like carbon	1.79 + 0.0099i [54]
Surrounding	Air	1.00002 [54]

.

2.4. Genetic Algorithm Optimization

To optimize the CPWR structures, the GA in MATLAB combined with the TMM approach was implemented. We began the algorithms by creating an initial population size of 50 solutions and then running these solutions through a Fitness Function (FF) that was given by:

FF = min(1/(o2 − o1))

(7)

where o2 and o1 were the reflection minima of the TM response at the higher and lower resonance angles, respectively.

The constraints were determined by the number of reflection minima (peak): peak ≤ 2, and by the sum of values of reflectivity of the waveguide resonance: R_max1 + R_min1 ≥ 0.9. Other parameters used were exponential scaling, randomizing mutation, and one-point crossover. For two reflection minima, the thicknesses of the TiN and AlN films were assumed to be variable in the range of [10, 100] nm and [1300, 5000] nm, respectively.

3. Results

3.1. The Results from the GA Approach

Table 2. The optimum of the TiN and AlN layer thicknesses of three CPWR sensors, predicted by GA method.

Structure	Thickness
	TiN Layer	AlN Layer
sapphire prism/TiN/AlN/air	29.42 nm	909.83 nm
magnesium oxide-prism/TiN/AlN/air	28.33 nm	909.50 nm
silicon prism/TiN/AlN/air	28.33 nm	771.88 nm

shows the results of utilizing the GA technique to predict the optimum TiN and AlN film thicknesses for three CPWR structures with air as the surrounding medium. From the results, the TiN film thickness for the magnesium oxide prism/TiN/AlN/air and silicon prism/TiN/AlN/air sensors was the same, which is a little thinner than the sapphire prism/TiN/AlN/air structure. For the silicon prism/TiN/AlN/air sensors, the GA method predicted that the thickness of the AlN layer was thinner than the sapphire prism/TiN/AlN/air and the magnesium oxide prism/TiN/AlN/air structures. The next part discusses the reflectivity curves as well as the TM and TE modes generated in the waveguides for the three structures.

3.2. Theoretical CPWR Spectra

Figure 2 shows the reflectivity curves as a function of incident angle for the three GA optimized CPWR structures in the ambient air. The reflectivity from the sapphire prism/29.42 nm TiN/ 909.83 nm AlN/air sensor displayed two minima for TM polarized incident light (35.91° and 76.50°) and a single minimum for TE-polarized incident light (56.97°). The reflectivity minima at the incident angles of 35.91° and 76.50° correspond to the waveguide resonance and the SPR, respectively. The reflectivity from the magnesium oxide prism/28.33 nm TiN/ 909.50 nm AlN/air sensor exhibited two minima for TM-polarized incident light (36.72° and 80.91°) and a single minimum for TE-polarized incident light (58.77°). The 36.72° and 80.91° minima correspond to the waveguide resonance and the SPR angles, respectively. The reflectivity from the silicon prism/28.33 nm TiN/ 711.88 nm AlN/air sensor showed two minima for TM-polarized incident light (25.02° and 36.99°) and two minimum for TE-polarized incident light (19.35° and 31.77°). The 25.02° and 36.99° minima correspond to the waveguide resonance and SPR angles, respectively.

3.3. Field Distribution for TM Polarized Light and Penetration Depth

The field intensity relative to the incident field $\left({\left|E_x\right|}^2/{\left|E_x^{inc}\right|}^2\right)$ for three optimized CPWR structures was simulated using the incident angles that correspond to the reflectivity minima (Figure 3). For an AlN thickness of 909.83 nm, the field profiles of the optimized sapphire prism/TiN/AlN/air sensor and magnesium oxide prism/TiN/AlN/air sensor were identical. At the SPR angles, the fields were enhanced by about 40-fold (Figure 3a) and 70-fold (Figure 3b) relative to the incident field for the optimized sapphire prism/TiN/AlN/air and magnesium oxide prism/TiN/AlN/air sensors, respectively. For the silicon prism/TiN/AlN/air sensor, the field at the prism/TiN interface increased as the usual surface plasmon field, which was only enhanced by about 5-fold, and there was a TM 3 mode in the AlN waveguide layer (Figure 3c).

An important parameter when working at a wavelength of 1500 nm is the penetration depth, which quantifies the extent of light penetration into the analyte medium, where the absolute value of the x component of the electric field (|Ex|) decays by factor of 1/e [55,56]. Figure 4 shows the calculated penetration depths at a wavelength of 1500 nm for TM–polarized incident light for each GA–optimized CPWR sensor. The penetration depth of the sapphire prism/TiN/AlN/air sensor was nearly equal to that of the magnesium oxide prism/TiN/AlN/air sensor and slightly higher than that of the silicon prism/TiN/AlN/air sensor.

3.4. Field Distribution for TE Polarized Light

The TE–polarized incident light is also beneficial since the TE modes may provide information about the anisotropy in ultrathin films. The detailed descriptions and examples for lipid orientation determination can be found in [57]. Figure 5 shows the magnetic field distribution for TE modes in the AlN layer for the optimized CPWR sensors. The TE modes in the AlN waveguides were the same for these three sensors. For the sapphire prism/TiN/AlN/air and magnesium oxide prism/TiN/AlN/air sensors, the magnetic fields were increased by around 23-fold and 20-fold, respectively, in comparison to the incident field. For the silicon prism/TiN/AlN/air sensor, the field enhancement factor was lower as compared with the sapphire prism/TiN/AlN/air and magnesium oxide prism/TiN/AlN/air sensors. However, the field was distributed at a distance beyond the AlN/air interface, which could adequately detect the DLC film deposited on top of the AlN layer.

3.5. The Optimized CPWR Sensors Responds to the DLC Film Thicknesses

We calculated the reflectivity spectra of the optimized CPWR sensors for TM-polarized (Figure 6) and TE-polarized incident light (Figure 7) to respond to the deposition of the DLC films. With increasing DLC film thickness from 1 to 10 nm in steps of 1 nm, the resonance angles for the TM and TE polarizations shifted to higher angles. Evidently, the silicon prism/TiN/AlN sensor was less sensitive to the DLC film deposited on top of the AlN layer. The thickness sensitivity of a sensor is defined as the ratio of the shift in the resonance angle Δθ to the variation in the thickness of a coating layer Δd. The sensitivities for all optimized CPWR sensors are shown in Figure 8. The shifts in resonance angle curves used for the thickness sensitivity calculation were taken from Figure 6 and Figure 7. The curves were linearly fitted with an average R-square of 0.9995. For TM-polarized incident light of 35.91° and 76.50°, the thickness sensitivity was 0.026 and 0.101, respectively, for the sapphire prism/TiN/AlN/air sensor (Figure 8a). For TM-polarized incident light of 36.72° and 80.91°, the thickness sensitivity was 0.019 and 0.143, respectively, for magnesium oxide prism/TiN/AlN/air sensor (Figure 8b). For TM-polarized incident light of 25.02° and 36.99°, the thickness sensitivity was 0.029 and 0.002, respectively, for silicon prism/TiN/AlN/air sensor (Figure 8c).

Figure 9 shows the calculated sensitivities for the optimized CPWR sensors as a function of the RI of the thin DLC films and clearly displays the increase linearly in sensitivity towards RI changes of the DLC film. The magnesium oxide prism/TiN/AlN sensor had the highest sensitivity compared to the sapphire prism/TiN/AlN and silicon prism/TiN/AlN sensors. The sensitivity of the SPR at an incident angle of 36.99° for the silicon prism/TiN/AlN sensors was nearly zero. We noted that the optimized sapphire prism/TiN/AlN sensor had a sensitivity comparable to the sensitivity of gold and silica dielectric-loaded waveguide spectroscopy [10].

3.6. Example for Simultaneous Measuring the Thickness and Refractive Index of the DLC Film Using the GA Optimized Sapphire Prism/TiN/AlN/Air Sensor

To get a sense of the experimental use of the optimized CPWR sensors to simultaneously measure the thickness and RI of DLC films, we showed how to extract the thickness and RI of thin DLC films using the graphical technique for the sapphire prism/29.42 nm TiN/909.83 nm AlN sensor (Figure 10). Two trial curves were created for two reflectivity minima for TM-polarized incident light (35.91° and 76.50° in Figure 2). One trial curve (dash) was created at the waveguide resonance angle (at 35.91°) using seven indices of refraction ranging from 1.74 to 1.89. Another trial curve (solid) was created at the surface plasmon resonance angle (at 76.50°) using seven indices of refraction ranging from 1.68 to 1.94. The intersection point of the trial curves was the unique solution for the DLC film thickness and the refractive index. For our example, the intersection point in Figure 10 was (9.14, 1.79), indicating that the thickness and the refractive index for the DLC film were 9.14 nm and 1.79, respectively.

4. Discussion

Many techniques, such as SPR and ellipsometry (SE), require at least two independent measurements to calculate the thickness and RI of thin films simultaneously. The two-color [58], multi-wavelength [59], two-medium [60], and two-thickness [61] techniques are all examples of SPR applications. The drawback of the two-color method, which used two colors of a light source, is that the dispersion function of the thin films must be known. The two-medium approach, using two different external mediums, involves the constraint that the thin films being measured have no interaction with the media. The disadvantage of the two-thickness technique is that prior information about the nominal thickness ratio of the two thin films must be available. In practice, the thickness and RI of thin films have been widely determined using the SE. The disadvantage of SE is that it is expensive and needs the use of elaborate theoretical models for data analysis [62]. Moreover, the SE has proven difficult to determine the thickness and RI of thin films with a thickness of less than 15 nm [63].

Table 3. The advantages and disadvantages of methods for simultaneous measurement of the thickness and RI of thin films.

Method		Advantage	Disadvantage
Conventional SPR: prism/metal/air	One wavelength [64,65]	- high sensitivity - simple for fabrication - sharp resonance	- thickness and refractive index cannot be determined from single measurement - incompatibility with CMOS fabrication processes
	Two-color [66] Three-wavelength [67]	- thickness and refractive index can be determined from single measurement	- dispersion function of the thin films must be known
	Two-medium [60]		- thin films being measured must have no interaction with the media
Optical fiber SPR: SiO2/Au/water	One wavelength [6]	- simple for fabrication - sharp resonance - compact size - cheap	- thickness and refractive index cannot be determined from single measurement
Conventional CPWR: prism/Au/SiO2/air	One wavelength [15,16,17,18]	- thickness and refractive index can be determined from single measurement	- poor thermal stability - incompatibility with CMOS fabrication processes
Proposed CPWR: prism/TiN/AlN/air	One wavelength	- thickness and refractive index can be determined from single measurement - compatibility with CMOS fabrication process	- wider and shallower resonance - more sensitive to noise
Ellipsometry [62,63]	Multi wavelength	- able to detect ultrathin layers	- required complicate model - samples must be prepared - slow and difficult - expensive
Interferometry [68]	One wavelength	- low detection limits - high accuracy	- large size, cumbersome - required system calibration

summarizes the advantages and disadvantages of the two-resonance minima method provided in this work in comparison to previous approaches.

We note that the two reflection minima for the silicon prism/TiN/AlN/air sensor had significantly different sensitivity to the DLC film deposited on top of the AlN layer. As a result, the silicon prism/TiN/AlN/air may be used for self-reference dual-mode thickness sensors [69].

Although the field distribution, penetration depth, and sensitivity of the designed sensors for the magnesium oxide prism were larger than those for the sapphire prism, the sapphires were important for producing optoelectronic devices.

Recent research has explored replacing noble metallic films in sensors based on SPR with alternative plasmonic materials having a high damage threshold and compatibility with standard CMOS technology. This may produce a low-cost, easy-to-use, and CMOS-compatible fabrication process for sensors. Furthermore, sensors that work in the NIR spectral range are important since lasers and detectors are cheap and easily available. Our work on the development of the CPWR sensors made of alternative materials in the NIR wavelength range would allow the further adoption of this class of materials and enable new sensing applications.

5. Conclusions

The designs of CPWR sensors composed of titanium nitride and aluminum nitride thin films for simultaneous thickness and RI measurements of DLC thin films in air at a wavelength of 1550 nm were explored. We optimized the design of CPWR sensors using the genetic algorithm. We examined the responses of the optimized CPWR sensors to the changes in reflectivity curves when the DLC thin films were deposited on top of the CPWR sensors. Furthermore, we simulated the electromagnetic field profiles using the TMM and the FDTD method to understand the coupling between photons and SPPs.

From the simulation results, we concluded that the optimized CPWR sensor could simultaneously characterize the thickness and the refractive index of the DLC films with a minimum thickness of 1.0 nm. The sensitivity of the optimized CPWR sensor was equivalent to that of the conventional CPWR devices. The experimental tests are needed to verify our design and optimization of the CPWR sensors. The design and optimization methods used in this study should be broadly applicable across other plasmon-waveguide resonance sensors.