Performance of Surface Plasmon Resonance Sensors Using Copper/Copper Oxide Films: Influence of Thicknesses and Optical Properties

Abstract

Surface plasmon resonance sensors (SPR) using copper for sensitive parts are a competitive alternative to gold and silver. Copper oxide is a semiconductor and has a non-toxic nature. The unavoidable presence of copper oxide may be of interest as it is non-toxic, but it modifies the condition of resonance and the performance of the sensor. Therefore, the characterization of the optical properties of copper and copper oxide thin films is of interest. We propose a method to recover both the thicknesses and optical properties of copper and copper oxide from absorbance curves over the $(0.9;3.5)$ eV range, and we use these results to numerically investigate the surface plasmon resonance of copper/copper oxide thin films. Samples of initial copper thicknesses 10, 30 and 50 nm, after nine successive oxidations, are systematically studied to simulate the signal of a Surface Plasmon Resonance setup. The results obtained from the resolution of the inverse problem of absorbance are used to discuss the performance of a copper-oxide sensor and, therefore, to evaluate the optimal thicknesses.

1. Introduction

Surface plasmon resonance sensors are widely used for their high-sensitivity in real-time detection. Many improvements of the basic SPR [1], using a single gold layer as the active part of the sensor, have been proposed to improve the sensor performance [2]. For example, adding a specific absentee layer of KCL or Si₃N₄ or an absentee porous silica film can improve the performance of the SPR sensor by 5%, 11% and more than 300% [3,4]. The performance of the SPR sensor can be evaluated by the depth of the resonance dip, the full width at half maximum (FWHM) of the dip, the sensitivity to changes of the refractive index of the medium of detection and the figure of merit (FOM). The figure of merit is the ratio of the sensitivity to the FWHM. Indeed, a smaller FWHM gives a higher signal-to-noise ratio and higher measurement accuracy [3].

Copper presents interesting properties for optics, nanotechnology [5,6,7] and, therefore, for SPR sensors [8,9,10]. A recent paper demonstrated that the performance of a Cu-SPR sensor is quite similar to Au-SPR ones [9]. Moreover, bovine serum albumin (BSA) and other proteins do not bind irreversibly onto thin Cu-films. Copper phthalocyanine Langmuir–Blodgett films have been characterized using surface plasmon resonance (SPR) [11]. Exposure to toluene resulted in a partially reversible shift in the resonance depth and position of the SPR curves [12]. Moreover, the high cost of silver and gold materials—widely used in nanotechnologies—makes copper an adequate alternative to them.

Copper layers are naturally oxidized and their oxidation can be controlled by annealing. Copper oxides have non-toxic nature. The performance of SPR sensors depends not only on the metal layer used but also on dielectric layers above. Therefore, the study of Cu-oxide-SPR sensor performance requires finding out the thickness and optical properties of both materials, optical properties of copper and copper oxide thin films being different from those of bulk materials [13]. Consequently, to investigate Cu-oxide-SPR sensors, we first determine simultaneously the thicknesses and optical properties of 30 bilayers, from the fitting of absorbance curves. We detail the characteristics of the samples (Appendix A and Appendix B), the methods used for fitting (Appendix C), and the results (Appendix C.4) in the Appendix. Then, we use the recovered data to analyze numerically and discuss the Cu-oxide-SPR sensor performance. We propose the same electromagnetic approach to simulate the SPR signal and the metric of SPR performance (Section 2.2). The calculated SPR signals and performances are given in Section 3 and discussed in Section 4. Two cases are considered: the dry case (the upper medium is air), in which the SPR is used to analyze molecules in air, and the wet case (the medium of detection is water), in which the SPR characterize analytes in solutions. The optimal thicknesses are deduced from the performance parameters.

2. Materials and Methods

First, we summarize the method used to simultaneously determine the thicknesses and optical properties from experimental absorbance curves. This is an opportunity to detail the electromagnetic model of the generalized Fresnel coefficients also used for modeling the SPR sensor. We also give insights into the performance of SPR sensors.

2.1. Determination of Thicknesses and Optical Properties

The investigated samples are thin copper layers of target deposition thicknesses 10, 30 and 50 nm. They are successively annealed to control their oxidation under specific temperatures and for a given time. Thirty samples are characterized through the measurement of absorbance. Appendix A provides details on the sample preparation and experimental characterization [14]. The methods we propose for absorbance fitting and results are detailed in Appendix C.1, Appendix C.2, Appendix C.3 and Appendix C.4 and summarized in the following.

In Reference [15], we targeted the best fitting of UV-visible-NIR absorbance curves by a multilayer electromagnetic model using a function describing the optical properties over the range of investigated photon energies ($(0.9;3.5)$ eV, i.e., $(350;138)$ nm). Our methodology relied on using a single metaheuristic optimization method and the Drude–Lorentz model for the optical properties of copper. Consequently, the inverse problem was solved, revealing the best parameters of the electromagnetic and materials models. In this paper, we improve the method proposed in [15] to recover the thicknesses and optical properties of thin layers of copper/copper oxide from absorbance measurements as follows.

• In the model of optical properties, we replace the Drude–Lorentz two-terms model of dispersion by a sum of $N_{PF}$ partial fraction functions for fitting copper and copper oxide properties as a function of the photon energy $\hslash \omega$ ($N_{PF}=4$ and $N_{PF}=2$, respectively). $N_{PF}$ is the order of the partial fractions model, also called the complex-conjugate pole-residue pair model [16]. $N_{PF}$ is actually the number of poles in the function (see Equation (5)).
• We replace the two-steps method using first the bulk properties by a single multi-objective method, involving the error of fit and the relative difference between the optical properties we found and the bulk material properties. The purpose is to decrease the computational time.
• To achieve the fitting, we use three metaheuristic methods: Particle Swarm Method (PSO), Genetic Algorithm (GA), and Artificial Bee Colony method (ABC), with Gradient (GR) or Nelder–Mead simplex (NM) hybridization. Actually, metaheuristics are more or less efficient for achieving optimization, depending on the dimension of the problem (the number of input parameters of the model that have to be found) and on the topology of the objective function: the error of fit. Moreover, many realizations of the metaheuristics must be analyzed carefully to numerically characterize the stability of the solution and, hopefully, its uniqueness.

The calculation of the absorbance A is based on the electromagnetic model of the normal transmission of light across two plane material layers (with refractive index and thicknesses denoted $n_{Cu}$ and $h_{Cu}$ for copper, $n_{ox}$, $h_{ox}$ for oxide). The copper layer is deposited on a dielectric glass substrate of refractive index $n_g$ (medium 1). The absorbance A is deduced from the transmitted intensity across the successive material layers according to:

$$A=-log\left(|t_{14}{|}^2{T}_{AG}\right),$$

(1)

where:

$$T_{AG}={\left|\frac{2n_g}{n_g+1}\right|}^2,$$

(2)

and

$$t_{14}=\left(8n_{Cu}{n}_{ox}\right)exp\left(\mathrm{i}{k}_0(h_{Cu}(n_{Cu}-n_g)+h_{ox}(n_{ox}-n_g))\right)/D_0.$$

(3)

$\mathrm{i}$ is the pure imaginary number, and $k_0$ is the magnitude of the illumination wave vector. The denominator $D_0$ is:

$$\begin{array}{ccc}\hfill D_0& =& (n_{Cu}+n_g)(n_{ox}+n_{Cu})(1+n_{ox})\hfill \\ \hfill & & +(n_{Cu}-n_g)(n_{ox}-n_{Cu})(1+n_{ox})exp\left(2\mathrm{i}{k}_0{h}_{Cu}{n}_{Cu}\right)\hfill \\ \hfill & & +(n_{Cu}-n_g)(n_{ox}+n_{Cu})(1-n_{ox})exp\left(2\mathrm{i}{k}_0(h_{Cu}{n}_{Cu}+h_{ox}{n}_{ox})\right)\hfill \\ \hfill & & +(n_{Cu}+n_g)(n_{ox}-n_{Cu})(1-n_{ox})exp\left(2\mathrm{i}{k}_0{h}_{ox}{n}_{ox}\right).\hfill \end{array}$$

(4)

This electromagnetic model of generalized Fresnel coefficients includes the transmission of light across the Air-Quartz substrate $T_{AG}$. The optical properties $n_g$ of the glass substrate are measured before its coating with copper. The optical properties of copper and copper oxide are modeled as follows.

We use the partial fraction model, also called the complex-conjugate pole-residue pair model [16], to describe the dispersion of materials [17,18]. In a recent paper, the partial fraction model of dispersion yielded more accurate results than a sum of Drude–Lorentz functions to fit the bulk copper [18]. The relative permittivity is a complex number that is the square of the refractive index n:

$$n^2=n_{\infty }^2+\sum _{m=1}^{N_{PF}}\frac{c_m}{\mathrm{i}\omega -p_m}+\frac{c_m^{*}}{\mathrm{i}\omega -p_m^{*}},$$

(5)

where $X^{*}$ denotes the complex conjugate of Xy, $N_{PF}$ is the model order (number of poles), $\omega$ is angular frequency of the incoming light (the photon energy is $E=\hslash \omega$). $n_{\infty }^2$ is the value of the limit of the permittivity when $\omega$ tends toward infinity and is actually the contribution of interband transitions. Therefore, it is spectrally located beyond the investigated energy range. $p_i$ are poles and $c_i$ are residues. The occurrence of poles in permittivity corresponds to light–matter damped resonances in the complex plane [19,20]. They can also be related to electronic transitions in materials. This partial fraction model is used for both copper and copper oxide. In this study, we use model order $N_{PF}=2$ for copper oxide (semiconductor) and $N_{PF}=4$ for copper (metal) to maintain sufficient accuracy on the optical properties.

Therefore, the number of unknowns of the problem is $D=28$ by splitting complex numbers into two real numbers: two thicknesses and $1+4N_{PF}$ parameters for each PF dispersion model ($N_{PF}=4$ for copper and $N_{PF}=2$ for copper oxide). The model of absorbance (Equation (1)), including the models of optical properties (Equation (5)), is used within an evolutionary loop to find the best set of parameters for the model of absorbance. For the fitting of experimental UV-visible-NIR absorbance curves, a multi-objective function should be minimized to find the best input parameters of the fitting model.

The recovered parameters of the partial fraction model of dispersion (Equation (5)) are used to calculate the optical properties of copper and oxide at the wavelength of the laser incident light source ($632.8$ nm) used by the SPR sensor.

2.2. The SPR Model

The SPR setup configuration (Figure 1) was modeled by the electromagnetic interaction of light with a plane multilayer [21,22]. The same approach has been used to study the influence of the functionalization layer [21] and adhesion layer for nanostructured Au-EPR [23,24]. The light source is considered as a monochromatic plane wave of wavelength ${\lambda }_0=632.8$ nm (photon energy $E=\hslash \omega =1.959$ eV). For simplicity, we choose the same refractive index for the SPR prism as that used for the glass supporting the copper/copper oxide sample.

The SPR detected signal R is a function of the incident angle of the plane wave from normal to the multilayer surface:

$$R\left(\theta \right)=|r_{14}{|}^2{T}_{AG}{T}_{GA},$$

(6)

where $T_{AG}$ and $T_{GA}$ are the transmitted intensities of the light incoming and outcoming from the SPR setup (see Equation (2)). The reflected amplitude is $r_{14}$:

$$\begin{array}{ccc}\hfill r_{14}& =& ((n_{Cu}^2{k}_g^{\perp }-n_g^2{k}_{Cu}^{\perp })(n_{ox}^2{k}_{Cu}^{\perp }+n_{Cu}^2{k}_{ox}^{\perp })(n_4^2{k}_{ox}^{\perp }+n_{ox}^2{k}_A^{\perp })+\hfill \\ \hfill & & (n_{ox}^2{k}_{Cu}^{\perp }-n_{Cu}^2{k}_{ox}^{\perp })(n_{Cu}^2{k}_g^{\perp }+n_g^2{k}_{Cu}^{\perp })(n_4^2{k}_{ox}^{\perp }+n_{ox}^2{k}_A^{\perp })\hfill \\ \hfill & & exp\left(2\mathrm{i}{h}_{Cu}{k}_{Cu}^{\perp }\right)+\hfill \\ \hfill & & (n_4^2{k}_{ox}^{\perp }-n_{ox}^2{k}_A^{\perp })(n_{Cu}^2{k}_g^{\perp }+n_g^2{k}_{Cu}^{\perp })(n_{ox}^2{k}_{Cu}^{\perp }+n_{Cu}^2{k}_{ox}^{\perp })\hfill \\ \hfill & & exp\left(2\mathrm{i}(h_{ox}{k}_{ox}^{\perp }+h_{Cu}{k}_{Cu}^{\perp })\right)+\hfill \\ \hfill & & (n_{Cu}^2{k}_g^{\perp }-n_g^2{k}_{Cu}^{\perp })(n_{ox}^2{k}_{Cu}^{\perp }-n_{Cu}^2{k}_{ox}^{\perp })(n_4^2{k}_{ox}^{\perp }-n_{ox}^2{k}_A^{\perp })\hfill \\ \hfill & & exp\left(2\mathrm{i}{h}_{ox}{k}_{ox}^{\perp }\right))/D_{\theta }.\hfill \end{array}$$

(7)

$k^{\perp }$ being the normal to the surface of multilayer component of the wave vector. In medium m, this component is $k_i^{\perp }=k_0{(n_m^2-n_g^2{sin}^2\left(\theta \right))}^{1/2}$, with $k_0=\frac{\omega }{c}$, c being the speed of light in vacuum. The refractive index of the detection medium of the SPR sensor is $n_4$. The denominator $D_{\theta }$ of the transmitted amplitude is:

$$\begin{array}{ccc}\hfill D_{\theta }& =& (n_{Cu}^2{k}_g^{\perp }+n_g^2{k}_{Cu}^{\perp })(n_{ox}^2{k}_{Cu}^{\perp }+n_{Cu}^2{k}_{ox}^{\perp })(n_4^2{k}_{ox}^{\perp }+n_{ox}^2{k}_A^{\perp })+\hfill \\ \hfill & & (n_{Cu}^2{k}_g^{\perp }-n_g^2{k}_{Cu}^{\perp })(n_{ox}^2{k}_{Cu}^{\perp }-n_{Cu}^2{k}_{ox}^{\perp })(n_4^2{k}_{ox}^{\perp }+n_{ox}^2{k}_A^{\perp })\hfill \\ \hfill & & exp\left(2\mathrm{i}{h}_{Cu}{k}_{Cu}^{\perp }\right)+\hfill \\ \hfill & & (n_{Cu}^2{k}_g^{\perp }-n_g^2{k}_{Cu}^{\perp })(n_4^2{k}_{ox}^{\perp }-n_{ox}^2{k}_A^{\perp })(n_{ox}^2{k}_{Cu}^{\perp }+n_{Cu}^2{k}_{ox}^{\perp })\hfill \\ & & exp\left(2\mathrm{i}(h_{Cu}{k}_{Cu}^{\perp }+h_{ox}{k}_{ox}^{\perp })\right)+\hfill \\ \hfill & & (n_{ox}^2{k}_{Cu}^{\perp }-n_{Cu}^2{k}_{ox}^{\perp })(n_4^2{k}_{ox}^{\perp }-n_{ox}^2{k}_A^{\perp })(n_{Cu}^2{k}_g^{\perp }+n_g^2{k}_{Cu}^{\perp })\hfill \\ \hfill & & exp\left(2\mathrm{i}{h}_{ox}{k}_{ox}^{\perp }\right).\hfill \end{array}$$

(8)

We obtain $D_{\theta }=D_0$ (Equation (4)) by considering normal incidence of the light ($\theta =0$) and $n_4=1$. However, Equation (4) is used for the fitting of experimental UV-visible-NIR absorbance, as it requires less computational time.

The basics of SPR are given in Reference [25]. In that reference, we gave the conditions of SPR excitation on an interface between two mediums and the corresponding formula, also used in [9]. Considering a plane interface between two media (one of them is a metal), the SPR angle can be easily evaluated, $n_i^2$ and $n_j^2$ being the relative permittivities of materials on both sides of the interface, and $n_g^2$ being that of the hemispherical glass substrate, and ℜ the real part:

$$\theta (min\left(R\right))=\Re \left\{arcsin\left(\frac{n_i{n}_j}{{(n_i^2+n_j^2)}^{1/2}}\frac{1}{n_g}\right)\right\}$$

(9)

We consider the angle interrogation mode of the SPR sensor [2,9,12]. At a given wavelength, in the Angular Interrogation Mode (AIM), with a changing refractive index of water (upper medium, analyte), the resonance angle shifts [4].

2.3. Performance of the SPR Sensor

The performance parameters of the SPR sensor are the sensitivity $S_{\theta }$, the full width at half maximum ($FWHM$) and the Figure of Merit ($FOM$) [4]. The resonance angle found at the minimum of reflectance ($\theta (min(R)$), and the depth of the resonance dip ($min\left(R\right)$) are also of interest to characterize SPR sensors. The performance of SPR can help to determine the optimal thicknesses of the bilayer.

2.3.1. Sensitivity

The sensitivity of the SPR sensor is the angular shift that is found at the minimum of reflectance $R\left(\theta \right)$ by varying the refractive index of the medium of detection. The angular sensitivity is, therefore:

$$S_{\theta }=\frac{\theta (min\left(R\left(n_4^{\prime }\right)\right))-\theta (min\left(R\left(n_4\right)\right))}{RIU},$$

(10)

where $RIU$ is the relative index of refraction unit: $2(n_4^{\prime }-n_4)/(n_4^{\prime }+n_4)$. The greater the sensitivity is, the better performance of SPR.

2.3.2. Full Width at Half Maximum

The full width at half maximum is:

$$FWHM=max\left[\theta \left(\frac{max\left(R\right(\theta \left)\right)+min\left(R\right(\theta \left)\right)}{2}\right)\right]-min\left[\theta \left(\frac{max\left(R\right(\theta \left)\right)+min\left(R\right(\theta \left)\right)}{2}\right)\right].$$

(11)

The thinner the resonance dip is, the higher is the signal to noise ratio of the SPR sensor [3].

2.3.3. Figure of Merit

Therefore, the figure of merit of the SPR sensor is deduced from both the sensitivity and the full width at half maximum of the reflectance (FWHM) [4]:

$$\mathrm{FOM}=\frac{S_{\theta }}{\mathrm{FWHM}}.$$

(12)

The thinner the wells in the reflectance (small value of FWHM), the more enhanced the FOM. The greater the sensitivity deduced from the shift of the minimum in reflectance by a change of optical index of the upper medium, the greater the FOM. However, the FOM does not take into account the value of the minimum in reflectance $min\left(R\right)$. This minimum should be as close as possible to zero to obtain high dynamic of detection.

2.4. Optimal Thicknesses

Interference conditions in the metal layer have been considered in Reference [26] to find the optimal thickness of a classical SPR sensor. This condition relies on the calculation of the copper thickness $h_o$ that verifies constructive interference at the copper–oxide interface, after two reflections on the interfaces [27]:

$$2k_{Cu}^{\perp }{h}_o=\Re \left(m2\pi -arg\left(\frac{1}{r_{g-Cu}{r}_{Cu-ox}}\right)\right),$$

(13)

with m an integer number and $r_{g-Cu}$ and $r_{Cu-ox}$ the Fresnel coefficients of reflection on each interface. The real part of $h_o$ is actually an approximation of the optimal thickness, by neglecting the finite thickness of oxide. However, this criterion can be extended by calculating the argument of the sum of the illumination field and of the reflected one $arg(1+r_{14})$ (Equation (7)) that should be as close as possible to 0. In this case, we obtain a minimum of reflectance (destructive interference between the illumination and the reflected wave). This minimum corresponds to a pole (complex number) of the generalized Fresnel coefficient (Equation (7)) [19,28].

The maximum of the FOM (Equation (12)), the maximum of sensitivity (Equation (10)), the minimum of FWHM (Equation (11)), or even the minimum of reflectance $min\left(R\right)$ can also reveal the best thicknesses of copper and copper-oxide. We discuss the results obtained for all these criterion in Section 4.

2.5. Investigated Samples

The manufacturing and characterization of the investigated samples are detailed in Appendix A and Appendix B. The fitting methods are described in Appendix C, and the full results are given in Appendix C.4 (thicknesses of the copper and oxide layers, optical properties $@632.8$ nm, $1.96$ eV) for each of the thirty investigated samples. We summarize the main results shown in

Table A3. The recovered thickness of copper and copper oxide, and the relative permittivity of copper and copper oxide at ${\lambda }_0=632.8$ nm, by fitting experimental UV-visible-NIR absorbance curves. The value of the multi-objective function F (Equation (A3)) and the most efficient metaheuristics are also indicated. Values between brackets are standard deviations for acceptable solutions, i.e., obtained for F below a 25% threshold (for $F\in [min(F);1.25\times min(F\left)\right]$. The corresponding success of each metaheuristic is indicated in percents).

Sample	${\mathit{h}}_{\mathit{Cu}}$	${\mathit{h}}_{\mathit{ox}}$	${\mathit{n}}_{\mathit{Cu}}^2$	${\mathit{n}}_{\mathit{ox}}^2$	F	Method
	(nm)	(nm)	at $\mathit{\hslash }\mathit{\omega }=1.96$ eV	at $\mathit{\hslash }\mathit{\omega }=1.96$ eV
A010nmCut0	9.8	1.8	−12.5 + 3.6$\mathrm{i}$	8.2 + 1.2$\mathrm{i}$	7.9× 10${}^{-3}$	EM + NM
	(0.2)	(1.1)	(0.5 + 0.5$\mathrm{i}$)	(0.4 + 0.5$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 37%, $n_{EM}=$ 63%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm22min120	9.7	2.8	−11.9 + 3.3$\mathrm{i}$	8.2 + 1.3$\mathrm{i}$	5.1 × 10${}^{-3}$	ABC + NM
	(0.2)	(1.2)	(0.5 + 0.5$\mathrm{i}$)	(0.4 + 0.5$\mathrm{i}$)	(1 × 10${}^{-3}$)
	$n_{ABC}=$ 100%, $n_{EM}=$ 0%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm72min120	9.6	2.8	−12.5 + 3.6$\mathrm{i}$	7.8 + 1.1$\mathrm{i}$	7.1 × 10${}^{-3}$	ABC + NM
	(0.3)	(1.4)	(0.7 + 0.5$\mathrm{i}$)	(0.5 + 0.4$\mathrm{i}$)	(8 × 10${}^{-4}$)
	$n_{ABC}=$ 43%, $n_{EM}=$ 57%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm102min120	9.5	3.2	−14.1 + 3.2$\mathrm{i}$	7.7 + 1.0$\mathrm{i}$	8.0 × 10${}^{-3}$	ABC + NM
	(0.3)	(1.5)	(0.8+0.5$\mathrm{i}$)	(0.5 + 0.4$\mathrm{i}$)	(7 × 10${}^{-4}$)
	$n_{ABC}=$ 100%, $n_{EM}=$ 0%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm160min120	8.7	4.9	−15.4 + 4.8$\mathrm{i}$	8.0 + 1.0$\mathrm{i}$	1.1 × 10${}^{-2}$	EM + NM
	(0.5)	(1.0)	(1.1 + 1.0$\mathrm{i}$)	(1.1 + 0.5$\mathrm{i}$)	(6 × 10${}^{-4}$)
	$n_{ABC}=$ 36%, $n_{EM}=$ 45%, $n_{PSO}=$ 19%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm190min120	8.5	6.9	−14.8 + 5.6$\mathrm{i}$	7.5 + 1.2$\mathrm{i}$	9.3 × 10${}^{-3}$	EM + NM
	(0.5)	(1.1)	(1.2 + 1.0$\mathrm{i}$)	(1.1 + 0.5$\mathrm{i}$)	(6 × 10${}^{-4}$)
	$n_{ABC}=$ 60%, $n_{EM}=$ 40%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm240min130	2.6	16.7	−11.1 + 3.4$\mathrm{i}$	9.6 + 0.6$\mathrm{i}$	6.2 × 10${}^{-3}$	EM + NM
	(0.7)	(1.1)	(0.4 + 0.6$\mathrm{i}$)	(0.3 + 0.2$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 24%, $n_{EM}=$ 29%, $n_{PSO}=$ 47%, $n_{NM}=$ 92%, $n_{GR}=$ 8
A010nm600min130	1.9	17.9	−11.2 + 3.2$\mathrm{i}$	8.6 + 0.1$\mathrm{i}$	7.6 × 10${}^{-3}$	ABC + NM
	(0.5)	(0.9)	(0.5 + 0.5$\mathrm{i}$)	(0.4 + 0.0$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 50%, $n_{EM}=$ 22%, $n_{PSO}=$ 28%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm900min170	1.2	18.5	−14.2 + 4.2$\mathrm{i}$	8.5 + 0.0$\mathrm{i}$	1.6 × 10${}^{-2}$	PSO + NM
	(0.5)	(1.5)	(1.1 + 1.5$\mathrm{i}$)	(0.7 + 0.1$\mathrm{i}$)	(5 × 10${}^{-3}$)
	$n_{ABC}=$ 25%, $n_{EM}=$ 44%, $n_{PSO}=$ 31%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A010nm910min170	0.7	18.7	−11.0+3.3$\mathrm{i}$	8.3 + 0.0$\mathrm{i}$	6.4 × 10${}^{-3}$	ABC + NM
	(0.5)	(1.0)	(0.6 + 0.6$\mathrm{i}$)	(0.4 + 0.0$\mathrm{i}$)	(8 × 10${}^{-4}$)
	$n_{ABC}=$ 56%, $n_{EM}=$ 16%, $n_{PSO}=$ 28%, $n_{NM}=$ 100%, $n_{GR}=$ 0

,

Table A4. The recovered thickness of copper and Cooper oxide, the relative permittivity of copper and copper oxide at ${\lambda }_0=632.8$ nm, by fitting experimental UV-visible-NIR absorbance curves. The recovered relative permittivities can be compared to those of bulk copper and copper oxide, resp. −11.55 + 1.57$\mathrm{i}$ and 8.64 + 0.64$\mathrm{i}$ [29]. The value of the multi-objective function F (Equation (A3)) and the most efficient metaheuristics are also indicated. Values between brackets are standard deviation for acceptable solutions, i.e., obtained for F below the 25% threshold (for $F\in [min(F);1.25\times min(F\left)\right]$. The corresponding success of each metaheuristic is indicated in percentages).

Sample	${\mathit{h}}_{\mathit{Cu}}$	${\mathit{h}}_{\mathit{ox}}$	${\mathit{n}}_{\mathit{Cu}}^2$	${\mathit{n}}_{\mathit{ox}}^2$	F	Method
	(nm)	(nm)	at $\mathit{\hslash }\mathit{\omega }=1.96$ eV	at $\mathit{\hslash }\mathit{\omega }=1.96$ eV
A030nmCut0	30.0	0.0	−14.1 + 2.7$\mathrm{i}$	8.3 + 1.2$\mathrm{i}$	8.7 × 10${}^{-3}$	ABC + NM
	(0.1)	(0.7)	(0.5 + 0.6$\mathrm{i}$)	(0.2 + 0.3$\mathrm{i}$)	(4 × 10${}^{-4}$)
	$n_{ABC}=$ 62%, $n_{EM}=$ 38%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm22min120	29.9	0.1	−13.3 + 2.8$\mathrm{i}$	8.6 + 0.7$\mathrm{i}$	7.9 × 10${}^{-3}$	ABC + NM
	(0.1)	(0.5)	(0.5 + 0.5$\mathrm{i}$)	(0.3 + 0.3$\mathrm{i}$)	(8 × 10${}^{-4}$)
	$n_{ABC}=$ 42%, $n_{EM}=$ 50%, $n_{PSO}=$ 8%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm72min120	29.5	1.0	−12.5 + 2.9$\mathrm{i}$	8.2 + 1.0$\mathrm{i}$	6.2 × 10${}^{-3}$	EM + NM
	(0.3)	(0.5)	(0.4 + 0.4$\mathrm{i}$)	(0.3 + 0.3$\mathrm{i}$)	(4 × 10${}^{-4}$)
	$n_{ABC}=$ 44%, $n_{EM}=$ 56%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm102min120	29.4	1.0	−13.0 + 2.0$\mathrm{i}$	8.1 + 1.1$\mathrm{i}$	5.2 × 10${}^{-3}$	EM + NM
	(0.4)	(0.7)	(0.5 + 0.3$\mathrm{i}$)	(0.3 + 0.2$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 14%, $n_{EM}=$ 86%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm160min120	28.2	3.1	−12.8 + 3.1$\mathrm{i}$	8.3 + 0.5$\mathrm{i}$	4.9 × 10${}^{-3}$	ABC + NM
	(0.6)	(1.0)	(0.5 + 0.5$\mathrm{i}$)	(0.3 + 0.3$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 17%, $n_{EM}=$ 83%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm190min120	27.9	3.5	−13.5 + 2.9$\mathrm{i}$	8.4 + 0.6$\mathrm{i}$	4.8 × 10${}^{-3}$	ABC + NM
	(0.7)	(1.2)	(0.5 + 0.5$\mathrm{i}$)	(0.3 + 0.3$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 50%, $n_{EM}=$ 50%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm240min130	21.2	14.5	−11.3 + 8.2$\mathrm{i}$	7.7 + 1.3$\mathrm{i}$	1.1 × 10${}^{-2}$	EM + NM
	(1.5)	(2.4)	(1.1 + 1.5$\mathrm{i}$)	(0.9 + 1.1$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 9%, $n_{EM}=$ 70%, $n_{PSO}=$ 21%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm600min130	20.1	16.6	−15.9 + 6.8$\mathrm{i}$	7.7 + 0.6$\mathrm{i}$	1.2 × 10${}^{-2}$	EM + NM
	(1.2)	(2.2)	(0.9 + 0.8$\mathrm{i}$)	(0.9 + 0.6$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 18%, $n_{EM}=$ 57%, $n_{PSO}=$ 25%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm900min170	0.6	44.8	−10.2 + 2.2$\mathrm{i}$	7.0 + 0.0$\mathrm{i}$	1.1 × 10${}^{-2}$	EM + NM
	(4.2)	(6.6)	(0.9 + 1.0$\mathrm{i}$)	(1.1 + 0.8$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 4%, $n_{EM}=$ 96%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A030nm910min170	0.4	45.8	−11.5 + 3.1$\mathrm{i}$	7.1 + 0.0$\mathrm{i}$	1.1 × 10${}^{-2}$	EM + NM
	(4.2)	(6.6)	(0.9 + 1.0$\mathrm{i}$)	(1.1 + 0.8$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 0%, $n_{EM}=$ 40%, $n_{PSO}=$ 60%, $n_{NM}=$ 100%, $n_{GR}=$ 0

and

Table A5. The recovered thickness of copper and copper oxide, and the relative permittivity of copper and copper oxide at ${\lambda }_0=632.8$ nm, by fitting experimental UV-visible-NIR absorbance curves. The recovered relative permittivities can be compared to those of bulk copper and copper oxide, resp. −11.55 + 1.57$\mathrm{i}$ and 8.64 + 0.64$\mathrm{i}$ [29]. The value of the multi-objective function F (Equation (A3)) and the most efficient metaheuristics are also indicated. Values between brackets are standard deviation for acceptable solutions, i.e., obtained for F below 25% threshold (for $F\in [min(F);1.25\times min(F\left)\right]$. The corresponding success of each metaheuristic is indicated in percentages).

Sample	${\mathit{h}}_{\mathit{Cu}}$	${\mathit{h}}_{\mathit{ox}}$	${\mathit{n}}_{\mathit{Cu}}^2$	${\mathit{n}}_{\mathit{ox}}^2$	F	Method
	(nm)	(nm)	at $\mathit{\hslash }\mathit{\omega }=1.96$ eV	at $\mathit{\hslash }\mathit{\omega }=1.96$ eV
A050nmCut0	50.0	0.1	−15.4 + 2.0$\mathrm{i}$	7.9 + 1.0$\mathrm{i}$	1.0 × 10${}^{-2}$	EM + NM
	(0.1)	(0.2)	(0.5 + 0.4$\mathrm{i}$)	(0.4 + 0.3$\mathrm{i}$)	(5 × 10${}^{-4}$)
	$n_{ABC}=$ 38%, $n_{EM}=$ 62%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm22min120	49.8	0.1	−14.6 + 1.8$\mathrm{i}$	8.2 + 0.8$\mathrm{i}$	9.3 × 10${}^{-3}$	EM + NM
	(0.1)	(0.2)	(0.4 + 0.4$\mathrm{i}$)	(0.3 + 0.3$\mathrm{i}$)	(9 × 10${}^{-4}$)
	$n_{ABC}=$ 26%, $n_{EM}=$ 48%, $n_{PSO}=$ 26%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm72min120	49.8	0.6	−14.2 + 2.2$\mathrm{i}$	8.3 + 0.9 $\mathrm{i}$	7.0 × 10${}^{-3}$	EM + NM
	(0.3)	(0.3)	(0.4 + 0.4$\mathrm{i}$)	(0.3 + 0.3$\mathrm{i}$)	(4 × 10${}^{-4}$)
	$n_{ABC}=$ 33%, $n_{EM}=$ 51%, $n_{PSO}=$ 15%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm102min120	49.8	0.6	−13.2 + 2.3$\mathrm{i}$	8.8 + 0.5$\mathrm{i}$	5.7 × 10${}^{-3}$	EM + NM
	(0.3)	(0.3)	(0.3 + 0.5$\mathrm{i}$)	(0.3 + 0.3$\mathrm{i}$)	(7 × 10${}^{-4}$)
	$n_{ABC}=$ 33%, $n_{EM}=$ 56%, $n_{PSO}=$ 11%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm160min120	49.2	1.6	−12.5 + 2.3$\mathrm{i}$	8.4 + 0.8$\mathrm{i}$	5.6 × 10${}^{-2}$	EM + NM
	(0.5)	(0.6)	(0.4 + 0.4$\mathrm{i}$)	(0.3 + 0.2$\mathrm{i}$)	(4 × 10${}^{-4}$)
	$n_{ABC}=$ 38%, $n_{EM}=$ 59%, $n_{PSO}=$ 3%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm190min120	49.1	1.8	−12.2 + 2.3$\mathrm{i}$	8.3 + 0.7$\mathrm{i}$	5.5 × 10${}^{-3}$	EM + NM
	(0.5)	(0.6)	(0.4 + 0.4$\mathrm{i}$)	(0.3 + 0.2$\mathrm{i}$)	(3 × 10${}^{-4}$)
	$n_{ABC}=$ 43%, $n_{EM}=$ 57%, $n_{PSO}=$ 0%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm240min130	42.1	14.1	−13.7 + 2.6$\mathrm{i}$	7.4 + 1.3$\mathrm{i}$	7.1 × 10${}^{-3}$	PSO + NM
	(3.0)	(5.5)	(0.8 + 0.7$\mathrm{i}$)	(0.4 + 0.4$\mathrm{i}$)	(9 × 10${}^{-4}$)
	$n_{ABC}=$ 0%, $n_{EM}=$ 38%, $n_{PSO}=$ 62%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm600min130	39.0	19.0	−14.4 + 3.5$\mathrm{i}$	8.1 + 1.4$\mathrm{i}$	8.4 × 10${}^{-3}$	PSO + NM
	(3.8)	(6.7)	(1.0 + 0.8$\mathrm{i}$)	(0.3 + 0.7$\mathrm{i}$)	(1 × 10${}^{-3}$)
	$n_{ABC}=$ 29%, $n_{EM}=$ 31%, $n_{PSO}=$ 40%, $n_{NM}=$ 97%, $n_{GR}=$ 3
A050nm900min170	0.4	81.9	−9.1 + 2.3$\mathrm{i}$	6.1 + 0.1$\mathrm{i}$	1.4 × 10${}^{-2}$	EM + NM
	(0.6)	(4.6)	(0.9 + 1.1$\mathrm{i}$)	(0.5 + 0.1$\mathrm{i}$)	(1 × 10${}^{-3}$)
	$n_{ABC}=$ 7%, $n_{EM}=$ 83%, $n_{PSO}=$ 11%, $n_{NM}=$ 100%, $n_{GR}=$ 0
A050nm910min170	0.0	89.6	−11.5 + 2.8$\mathrm{i}$	6.6 + 0.3$\mathrm{i}$	1.4 × 10${}^{-2}$	ABC + NM
	(0.6)	(4.4)	(0.8 + 1.1$\mathrm{i}$)	(0.5 + 0.1$\mathrm{i}$)	(8 × 10${}^{-4}$)
	$n_{ABC}=$ 97%, $n_{EM}=$ 0%, $n_{PSO}=$ 3%, $n_{NM}=$ 100%, $n_{GR}=$ 0

:

• Copper and copper oxide thicknesses are globally in agreement with those obtained from experimental measurements (see Appendix B).
• The real part of the relative permittivity of copper is smaller than the bulk one for almost all samples. The imaginary part of the relative permittivity of copper is about twice the bulk one. The mean value of $n_{Cu}^2$ over the samples with $h_{Cu}>2$ nm is $-13.3+3.3\mathrm{i}$ (standard deviation $1.4+1.5\mathrm{i}$) compared to $n_{Cu}^2\left(bulk\right)=-11.6+1.6\mathrm{i}$ [29].
• The real part and the imaginary parts of the relative permittivity of oxide are close to that of the bulk, except for full oxidized samples for which both parts decrease. This behavior is probably due to air inclusion in oxide (the grain size of oxide can reach 80 nm, see Appendix B). The mean value of $n_{ox}^2$ over the samples with $h_{ox}>2$ nm is $8.2+1.0\mathrm{i}$ (standard deviation $0.4+0.3\mathrm{i}$) compared to $n_{\begin{array}{c}\hfill \mathrm{C}\mathrm{u}2\mathrm{O}\end{array}}^2\left(bulk\right)=8.6+0.6\mathrm{i}$ and $n_{\begin{array}{c}\hfill \mathrm{C}\mathrm{u}\mathrm{O}\end{array}}^2\left(bulk\right)=7.1+2.3\mathrm{i}$ [29]. If we suppose a chemical mix of both oxides, we deduce that oxide may be made of 76% Cu₂O and 24% CuO. This result was confirmed by XPS measurements [14,30]. The decrease in the real part of the oxide permittivity may also be due to air inclusion in oxide.
• For samples with roughness varying from 2 to 14 nm [14], the electromagnetic model of generalized Fresnel coefficients could be accurate enough. The quality of absorbance fitting shown in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10 and Figure A11 confirms the validity of the model, which can therefore be used to model the SPR.

3. Results

In this section, we use the thicknesses and optical properties that are simultaneously recovered from the fitting of absorbance curves (Table A3, Table A4 and Table A5) to study the SPR sensor setup, of which the sensitive part is a copper and copper oxide bilayer. From these results, we calculate the signal of the SPR sensor working in angular interrogation mode for air and water as the upper medium: in the dry and wet cases, respectively. We also evaluate the performance of such set up. We use $RIU=0.01$.

3.1. Dry Case

Figure 1 illustrates the SPR setup. The wavelength of laser illumination is ${\lambda }_0=632.8$ nm. The photon energy of the excitation light ($E=1.96$ eV) is close to that of the transition from d states (valence band) to th es-p conduction band [18] (2.1 eV, see the vertical black line in Figure A1, Figure A6 and Figure A11). Therefore, we expect a good quality of the plasmon resonance for adequate thickness of copper: a sharp dip and a small minimum.

Figure 2, Figure 3 and Figure 4 show the simulation of the SPR setup signal from the model of reflectance in Equation (6), considering air as the upper medium. The reflectance curves are plotted as functions of the incident angle of illumination (at ${\lambda }_0=632.8$ nm) for each investigated sample.

In Figure 2, for negligible thicknesses of copper (less than 2 nm), the reflectance is characteristic of a dielectric material. In the other cases, the wells in reflectance on the right correspond to the absorption of photon energy by the copper layer (resonance). Nevertheless, the quality of the surface plasmon resonance is low: the wells are wide and the depths of dip are greater than 0.26.

The surface plasmon dip for copper thicknesses close to 30 nm are thinner than in the previous case (Figure 3 vs. Figure 2). The smallest values of R (depth of dip) are close to each other for $h_{Cu}$ nm near 30 nm. The SPR angles are close to ${46}^{°}$. This SPR angle is close to that of a copper–air interface. The influence of the thin layer of oxide on the SPR angle is negligible for oxide thicknesses below 3.5 nm. However, the shape of the curve seems to be clearly modified, even for small thicknesses of the oxide. Thus, we can anticipate that the thickness of oxide will have an influence on the FWMH. For thicker oxide layers, the dip is widened and the SPR resonance angle approaches ${65}^{°}$, which is not so far from that of the copper–oxide interface (${72}^{°}$).

The SPR dips are thinner for the six samples for which $h_{Cu}\approx 50$ nm and are smaller than that for the previous samples (Figure 4 compared to Figure 2 and Figure 3). By decreasing $e_{Cu}$, the SPR dips are enlarged and shifted, as in the previous case. The performance parameters introduced above can be evaluated for the investigated samples.

The best performances of the SPR using the investigated bilayers as sensitive parts are shown in

Table 1. Calculated performance of the SPR setup given the retrieved thicknesses and optical properties in the dry case: resonance angle, the depth of SPR dip, full width at half maximum, sensitivity and figure of merit (only FOM $>20$ RIU${}^{-1}$ are indicated) and its uncertainty, FOM for optical properties of bulk materials, and for bare copper (without oxide).

${\mathit{h}}_{\mathit{Cu}}$	${\mathit{n}}_{\mathit{Cu}}^2$	${\mathbf{\theta }}_{\mathit{SPR}}$	$\mathbf{min}\left(\mathit{R}\right)$	FWHM	${\mathit{S}}_{\mathbf{\theta }}$	FOM	FOM	FOM
${\mathit{h}}_{\mathit{ox}}$	${\mathit{n}}_{\mathit{ox}}^2$						Bulk	ox. Free
(nm)		(${}^{°}$)		${}^{°}$	(${}^{°}$/RIU)	(RIU${}^{-1}$)	(RIU${}^{-1}$)	(RIU${}^{-1}$)
49.1	−12.2 + 2.3$\mathrm{i}$
1.8	8.3 + 0.7$\mathrm{i}$	46.8	0.08	3.2	67.5	20.9 (0.0)	27.8	25.8
49.2	−12. + 2.3$\mathrm{i}$
1.6	8.4 + 0.8$\mathrm{i}$	46.7	0.08	2.8	67.0	23.5 (0.0)	28.6	28.5
49.8	−13.2 + 2.3$\mathrm{i}$
0.6	8.8 + 0.5$\mathrm{i}$	46.2	0.09	2.1	65.0	31.2 (5.5)	32.9	33.5
49.8	−14. + 2.2$\mathrm{i}$
0.6	8.3 + 0.9$\mathrm{i}$	46.0	0.10	1.5	63.9	41.4 (0.0)	33.1	44.2
49.8	−14.6 + 1.8$\mathrm{i}$
0.1	8.2 + 0.8$\mathrm{i}$	45.8	0.07	1.1	63.4	57.3 (0.5)	34.9	57.7
50.0	−15.4 + 2.0$\mathrm{i}$
0.1	7.9 + 1.0$\mathrm{i}$	45.6	0.11	1.1	62.7	58.5 (0.0)	35.1	59.3

[3,4]. We indicate both the retrieved thicknesses of copper and oxide ($h_{Cu}$, $h_{ox}$) from absorbance curves, and the corresponding optical properties ($n_{Cu}^2$, $n_{ox}^2$) $632.8$ nm (extracted from Table A5). We give the resonance angle ${\theta }_{SPR}=\theta (min\left(R\right))$, the depth of dip $min\left(R\right)$, the full width at half maximum of the SPR dip (FWHM, Equation (11)), the sensitivity $S_{\theta }$ (Equation (10)), and the figure of merit (FOM, Equation (12)). The uncertainty on FOM is indicated in between brackets. This is the standard deviation of FOR for the recovered parameters of the multi-objective function $F\in [min(F);1.25\times min(F\left)\right]$ (see Appendix C). We also give the FOM calculated with the bulk optical properties for Cu and Cu₂O [29] and without the oxide layer for comparison.

The resonance angle ${\theta }_{SPR}$ is close to that of the interface between copper and the medium of detection (Equation (9), $45.3^{°}$). The minimum of reflectance is smaller than $0.1$ and the sensitivity is about the same for all samples. The FWHM is smaller for copper thicknesses close to 50 nm; therefore, the value of FOM is greater. The uncertainty on FOM is small for almost all cases. The thickness of oxide being negligible, the FOM for copper without oxide is about the same. The values of $S_{\theta }$ are greater than 62.7. On the contrary, the FOM calculated from bulk optical properties of copper and copper oxide are different. This is due to the value of the real part of the copper relative permittivity, which is smaller than that of bulk copper ($n_{Cu}^2\left(bulk\right)=-11.6+1.6\mathrm{i}$). This confirms the interest in simultaneously measuring the optical properties of copper and oxide for thin films.

In the general case, for a given thickness of copper and increasing thickness of oxide, the angle of resonance is shifted to the right, the FWHM increases as well as the depth of the SPR dip, and the FOM decreases.

The results for the investigated samples show that plasmon resonance can be launched for specific thicknesses. Therefore, the performance of the copper/copper oxide SPR setup deserves to be specifically studied, assuming that the medium of detection is water (wet case).

3.2. Wet Case

Table 2. Calculated performance of the SPR setup given the retrieved thicknesses and optical properties in the wet case: resonance angle, the depth of SPR dip, full width at half maximum, sensitivity and figure of merit (only FOM $>20$ RIU${}^{-1}$ are indicated) and its uncertainty, FOM for optical properties of bulk materials, and for bare copper (without oxide).

${\mathit{h}}_{\mathit{Cu}}$	${\mathit{n}}_{\mathit{Cu}}^2$	${\mathbf{\theta }}_{\mathit{SPR}}$	$\mathbf{min}\left(\mathit{R}\right)$	FWHM	${\mathit{S}}_{\mathbf{\theta }}$	FOM	FOM	FOM
${\mathit{h}}_{\mathit{ox}}$	${\mathit{n}}_{\mathit{ox}}^2$						Bulk	ox. Free
(nm)		(${}^{°}$)		${}^{°}$	(${}^{°}$/RIU)	(RIU${}^{-1}$)	(RIU${}^{-1}$)	(RIU${}^{-1}$)
27.9	−13.5 + 2.9$\mathrm{i}$
3.5	8.4 + 0.6$\mathrm{i}$	81.4	0.04	11.1	258.0	23.3 (1.1)	9.5	19.1
28.2	−12.8 + 3.1$\mathrm{i}$
3.1	8.3 + 0.5$\mathrm{i}$	81.4	0.03	11.3	253.0	22.4 (1.0)	10.1	18.7
29.4	−13.0 + 2.0$\mathrm{i}$
1.0	8.1 + 1.1$\mathrm{i}$	78.6	0.11	10.0	231.4	23.1 (1.0)	5.8	22.1
29.5	−12.5 + 2.9$\mathrm{i}$
1.0	8.2 + 1.0$\mathrm{i}$	78.9	0.04	10.9	229.6	21.1 (0.0)	5.9	20.0
29.9	−13.3 + 2.8$\mathrm{i}$
0.1	8.6 + 0.7$\mathrm{i}$	77.3	0.06	10.3	214.5	20.8 (0.2)	10.1	20.8
30.0	−14.1 + 2.7$\mathrm{i}$
0.0	8.3 + 1.2$\mathrm{i}$	76.8	0.06	9.9	210.7	21.4 (0.0)	5.9	21.3
49.1	−12.2 + 2.3$\mathrm{i}$
1.8	8.3 + 0.7$\mathrm{i}$	83.3	0.26	8.7	200.2	22.9 (0.0)	11.9	34.4
49.2	−12.5 + 2.3$\mathrm{i}$
1.6	8.4 + 0.8$\mathrm{i}$	82.8	0.23	8.6	252.8	29.2 (0.0)	16.4	35.5
49.8	−13.2 + 2.3$\mathrm{i}$
0.6	8.8 + 0.5$\mathrm{i}$	80.3	0.15	8.3	305.7	36.8 (2.1)	38.7	36.5
49.8	−13.2 + 2.3$\mathrm{i}$
0.6	8.8 + 0.5$\mathrm{i}$	80.3	0.15	8.3	305.7	36.8 (2.1)	38.7	36.5
49.8	−14.2 + 2.2$\mathrm{i}$
0.6	8.3 + 0.9$\mathrm{i}$	78.9	0.13	7.3	292.6	40.0 (0.0)	39.4	40.2
49.8	−14.6 + 1.8$\mathrm{i}$
0.1	8.2 + 0.8$\mathrm{i}$	77.8	0.08	5.9	277.7	47.2 (0.6)	43.7	47.3
50.0	−15.4 + 2.0$\mathrm{i}$
0.1	7.9+1.0$\mathrm{i}$	77.0	0.11	5.7	262.2	45.7 (0.0)	43.7	45.8

restates the best thicknesses of copper and oxide, the retrieved relative permittivity ($n_{Cu}^2$ and $n_{ox}^2$) extracted from Table A4 and Table A5 and the calculated performance of SPR for samples with FOM > 20.

The prism-SPR performance is evaluated as in the dry case. For water as upper medium, the SPR angle is greater than for air. The SPR angle falls between the SPR of the copper–oxide interface (72.4${}^{°}$) and that of the glass–copper interface (86.7${}^{°}$). On the contrary of the dry case, the samples with copper thicknesses around 28 and 30 nm give a reflectance smaller than 0.11 and FOM greater than 20. In this case, the values of both the sensitivity and the figure of merit are about the same whatever the oxide thickness is. The maximum of FOM is reached for thicker copper layers (near 50 nm), as in the dry case. The figure of merit is also calculated from the best values but by neglecting the oxide layer. In this case, we suppose that the bare copper layer is directly in contact with the above medium. Let us emphasize that the presence of oxide decreases the FOM (see FOM and FOM ox. free). This result is in agreement with that obtained for gold-SPR coated with porous silica film [4]. The performance of SPR working in dry and wet cases depends strongly on the thickness of both layers. The maximum sensitivity is obtained for the optimal thicknesses of 28 and 3 nm for copper and oxide, respectively, and for 49.8 and 0.6 nm.

4. Discussion

4.1. Cu-Oxide-SPR Performance

The performance of Cu-oxide-SPR in dry and wet cases are of the same order of magnitude as that mentioned in Reference [4] for silver/porous silica (wet case). The slight decrease in FOM by using a dielectric coating of metal can also be observed for our samples, for which the thickness of oxide is lower than 3.5 nm. The best values of FOM are obtained for oxide thicknesses lower than 3 nm. We observe that the FOM is highly sensitive to changes in the refractive index of copper. The FOM value is slightly greater than that obtained with Au, Au/Si₃N₄, Au/KCl materials [3]. The values of sensitivity and FWHM are greater in the wet case than in the dry case, leading to a smaller figure of merit.

In addition to its non-toxic nature, the oxide layer can be used to tune the position of the SPR resonance [9]. Indeed, the oxide layer could be used to adjust the resonance near the copper transition 2.1 eV that is characteristic of the inter-band transition from d states (valence band) to the ‘s-p’ conduction band [18] (Figure A1, Figure A6 and Figure A11 where the vertical line shows the photon energy of illumination used in the SPR setup $E=1.96$ eV). Let us note that cuprous copper oxide (Cu₂O) and cupric copper oxide (CuO) are p-type semiconductors with a bandgap of approximately 2.2–2.9 and 1.2–2.1 eV, respectively.

The optimal copper thickness (around 50 nm) is coherent with that found in Reference [9] ($45\pm 5$ nm), even if the substrate (prism) and the wavelength were not the same. However, these results show that more than one copper thickness may be used to design an efficient SPR sensor. This is the consequence of the high dependence of the optical properties on both the thickness and the sample elaboration mode (temperature and annealing time).

The SPR angle is around 46${}^{°}$ in the dry case and 80${}^{°}$ in the wet case. This last value is close to those obtained with a Al₂O₃ coating [10], and with BK7 substrate [9], both in the wet case. These SPR angles can also be compared to the SPR angles of single interfaces (Equation (9)). In the dry case, the shift of SPR angle is about 1–2${}^{°}$ for oxide thicknesses below 2 nm. In the wet case, the shift is greater and falls between that of the glass/copper and copper/water interfaces. Therefore, the coupling of both SPR differs for two different mediums of detection.

Therefore, the performance parameters may reveal different properties of the SPR-sensor. In the next section, we discuss the optimal thickness obtained from the best performance parameters.

4.2. Optimal Thicknesses

The thicknesses of copper for the investigated samples are not sufficient to determine the optimal thickness. Nevertheless, the small dispersion of retrieved optical properties of copper and oxide leads us to use their mean value to determine the optimal thicknesses of copper and oxide layers. Actually, the mean value is $n_{Cu}^2=-13.3+3.3\mathrm{i}$ (standard deviation $1.4+1.5\mathrm{i}$) and $n_{ox}^2=8.2+1.0\mathrm{i}$ (standard deviation $0.4+0.3\mathrm{i}$), see Section 2. They differ from the bulk values [29]. Therefore, we can use the mean values of the optical properties and a scan of copper thicknesses from 10 to 70 nm, and oxide thickness in (1;60) nm, in steps of 0.5 nm, to evaluate the optimal thicknesses from the best performance parameters and interference conditions, as explained in Section 2.4, in the dry and wet case.

Table 3. Mean optimal thicknesses for the top 1% performance parameters of the SPR set up in the dry case according to the minimum of reflectance, sensitivity, full width at half maximum, figure of merit and argument of the total field on the glass-copper interface. In brackets, standard deviation of the top 1%.

${\mathit{h}}_{\mathit{Cu}}$	${\mathit{h}}_{\mathit{ox}}$	$min\left(\mathit{R}\right)$	${\mathit{S}}_{\mathit{\theta }}$	$\mathit{FWHM}$	$\mathit{FOM}$	$arg(1+{\mathit{r}}_{14})$
(nm)	(nm)		${}^{°}$/${\mathbf{RIU}}^{-1}$	${}^{°}$	${\mathbf{RIU}}^{-1}$	${}^{°}$
Top 1% $min\left(R\right)$
28.1 (5.6)	14.9 (9.5)	0.003 (0.003)	73.7 (26.2)	23.7 (6.3)	3.3 (1.5)	3.1 (1.8)
Top 1% $min\left(\right|D\left|\right)$
31.9 (11.6)	51.0 (5.9)	1 (0)	0.0 (0.0)	90.0 (0.0)	0.0 (0.0)	0.0 (0.0)
Top 1% $min\left(\right|arg(1+r_{14})\left|\right)$
31.9 (11.5)	50.3 (7.7)	1 (0.1)	1.4 (10.7)	88.9 (8.5)	0.1 (0.5)	0.0 (0.0)
Top 1% $max\left(FWHM\right)$
46.0 (3.9)	1.0 (0.0)	0.1 (0.05)	65.4 (0.6)	5.5 (0.3)	12.0 (0.5)	14.5 (3.2)
Top 1% $max\left(FOM\right)$
46.2 (1.7)	1.0 (0.0)	0.1 (0.02)	65.2 (0.6)	5.2 (0.1)	12.5 (0.0)	14.9 (1.4)
Top 1% $max\left(S_{\theta }\right)$
54.1 (4.0)	16.9 (0.3)	0.3 (0.05)	125.0 (0.5)	30.1 (0.6)	4.2 (0.1)	30.9 (2.3)

and

Table 4. Optimal thicknesses for the top 1% performance parameters of the SPR set up in the wet case according to the minimum of reflectance, sensitivity, full width at half maximum, figure of merit and argument of the total field on the glass–copper interface. In brackets, standard deviation of the top 1%.

${\mathit{h}}_{\mathit{Cu}}$	${\mathit{h}}_{\mathit{ox}}$	$min\left(\mathit{R}\right)$	${\mathit{S}}_{\mathit{\theta }}$	$\mathit{FWHM}$	$\mathit{FOM}$	$arg(1+{\mathit{r}}_{14})$
(nm)	(nm)		${}^{°}$/${\mathbf{RIU}}^{-1}$	${}^{°}$	${\mathbf{RIU}}^{-1}$	${}^{°}$
Top 1% $min\left(R\right)$
24.8 (7.2)	6.5 (4.9)	0.004 (0.003)	199.5 (62.8)	10.7 (1.4)	18.2 (4.0)	3.6 (2.0)
Top 1% $min\left(\right|arg(1+r_{14})\left|\right)$
31.9 (11.5)	50.4 (7.4)	1 (0.06)	0.9 (14.8)	89.7 (4.7)	0.1 (1.3)	0.0 (0.0)
Top 1% $min\left(\right|D\left|\right)$
42.7 (17.0)	35.0 (15.3)	1 (0)	0.0 (0.0)	90.0 (0.0)	0.0 (0.0)	24.5 (18.2)
Top 1% $max\left(FWHM\right)$
10.9 (0.9)	17.9 (1.3)	0.03 (0.03)	49.3 (30.8)	7.4 (0.3)	6.7 (4.2)	9.8 (6.5)
Top 1% $max\left(FOM\right)$
43.5 (1.9)	1.0 (0.0)	0.1 (0.04)	265.5 (0.8)	11.0 (0.0)	24.1 (0.1)	27.3 (4.4)
Top 1% $max\left(S_{\theta }\right)$
44.0 (2.2)	1.0 (0.0)	0.1 (0.05)	265.4 (0.8)	11.0 (0.0)	24.0 (0.1)	28.4 (5.0)

give the mean value and the standard deviation of the top 1% results for each performance parameter.

The argument of $1+r_{14}$ should be close to 0 for these optimal values. This means that the conditions of destructive interference between incident and reflected waves are almost fulfilled and the minimum of reflectance is reached. The optimal thickness can be found for the maximum of FOM (Equation (12)) or sensitivity (Equation (10)), or for the minimum of FWHM and reflectance $min\left(R\right)$.

In the dry case, the best thicknesses of copper are close together for $min\left(R\right)$ and $min\left(\right|arg(1+r_{14})\left|\right)$, on the one hand, and for $max\left(FWHM\right)$, $max\left(FOM\right)$ and $max\left(S_{\theta }\right)$ on the other hand. If the conditions of constructive interference in the copper layer are fulfilled (Equation (13)), the optimal thicknesses of copper found are 29 nm for $m=5$, 48 nm for $m=8$. Therefore, we can deduce that all the copper thicknesses in Table 3 correspond to optimal ones, but for different optimal parameters. Moreover, these values are in agreement with that obtained from investigated samples in Section 3. The optimal thickness of copper for $max\left(FOM\right)$ is between those for $max\left(FWHM\right)$ and $max\left(S_{\theta }\right)$. Therefore, the optimal thicknesses can be deduced from a selection of the solution with the minimum of $min\left(R\right)$ among the top 1% solutions for $max\left(FOM\right)$: $e_{Cu}=44$ nm, $e_{ox}=1$ nm, $min\left(R\right)=0.09$, $FOM=12/,RI{U}^{-1}$, $FWHM=5^{°}$${\theta }_{SPR}=46.7^{°}$.

In the wet case, the best thicknesses of copper are close together for $min\left(R\right)$ and $min\left(\right|arg(1+r_{14})\left|\right)$, on the one hand, and for $max\left(FOM\right)$ and $max\left(S_{\theta }\right)$ on the other hand. If the conditions of constructive interference in the copper layer are fulfilled (Equation (13)), the optimal thicknesses of copper found are 29 nm for $m=5$, 42 nm for $m=7$. Again, we can deduce that all the copper thicknesses in Table 3 correspond to optimal ones, but for different optimal parameters. Furthermore, these results are also in agreement with that obtained from investigated samples in Section 3. The optimal thickness of copper for $max\left(FOM\right)$ is close to that for the $max\left(S_{\theta }\right)$ performance parameter. As in the dry case, the optimal thicknesses can be deduced from selection of the solution with the minimum of $min\left(R\right)$ among the top 1% solutions for $max\left(FOM\right)$: $e_{Cu}=41$ nm, $e_{ox}=1$ nm, $min\left(R\right)=0.09$, $FOM=24/,RI{U}^{-1}$, $FWHM={11}^{°}$${\theta }_{SPR}=86.9^{°}$.

5. Conclusions

The performance of Cu-oxide-SPR has been studied, including the influence of copper oxide. To reach this goal, we have used the fitting of experimental UV-visible-NIR absorbance curves to simultaneously obtain the thicknesses and optical properties of copper/copper oxide samples. For this, more than one metaheuristic is of interest. We found that the optical properties of copper and copper oxide vary as a function of the thicknesses and differ from the bulk ones. The performance of Cu-oxide-SPR confirms that copper/copper oxide could be a valuable alternative to gold for SPR sensors. The performance parameters reveal different optimal thicknesses in the dry and wet cases. For a systematic study of the optimal thickness, the combination of the maximum of FOM and the minimum of reflectance could be an interesting alternative to find the optimal thickness. The relevance of the result depends on the accurate determination of thicknesses and optical properties of the bilayer. At $\lambda =632.8$ nm, with $n_{Cu}^2=-13.3+3.3\mathrm{i}$, $n_{ox}^2=-8.2+1.0\mathrm{i}$, the optimal thicknesses of copper and oxide layers are about 44 and 1 nm in the dry case, and 41 and 1 nm in the wet case. They are determined to maximize the FOM and to minimize the dip magnitude. To optimize the SPR setup, we suggest the following process.

• Production series of samples with controlled annealing under low temperature;
• Characterization of samples (measurement of thickness and optical properties) from absorbance curves, for example;
• Characterization of samples in angular interaction mode;
• Fit of the SPR signal to verify the optical properties;
• Selection of the best sample and verification at regular time intervals that the thicknesses and optical properties remain the same.

Actually, the method of fit proposed in this paper could be applied to an SPR signal. In the future, we also intend to apply (and adapt if necessary) our method to characterize SPR with a multilayer sensitive part.