**4.1. The LSPR** *λ*0(*LSPR*)(*D*, *h*)

For both thicknesses of the chromium adhesion layer *e* = 2 and *e* = 6 nm, the position of LSPR is about the same when both *D* and *h* are varied respectively by 20 nm and 10 nm, with *P* = 200 nm, and a roughness of RMS= 1.6 nm. The resulting behavior law can be deduced from the parametric study: a steady position of the LSPR is observed, if the following law between *D* (nm) and *h* (nm) is satisfied:

$$D(P = 200, e = 2 - 6) = ah + b \approx (0.0084 \lambda\_0 (LSPR) - 3.86)h + (0.319 \lambda\_0 (LSPR) - 160) \tag{6}$$

This law is deduced from the least square fit of the results of the parametric study, simply by computing the slope and the intercept of *D* as a function of *h* for each LSPR position, and then by finding the linear dependence of *a* and *b* on the LSPR position *λ*0(*LSPR*). A similar approach was used to characterize the near-field optical microscopes and the evanescent near-field around nanostructures [1, 4]. The method uses a robust fit of the results of the parametric study. The algorithm uses iteratively reweighted least squares with the bisquare weighting function (Matlab).

Figure 7 shows the plots of Eq. 6, superimposed on figures 4. The linear behavior of *D* as a function of *h* is dependent on the LSPR position. Equation 6 helps to determine the geometrical parameters (*D* and *h*) of the biosensor to adjust the LSPR to a given wavelength.

**Figure 7.** Plots of *D* as a function of *h* for *λ*0(*LSPR*) ∈ {600; 640; 680; 720; 760; 800} from Eq. 6.

Similarly, the position of the LSPR (nm) can be deduced from Eq. 6 as a function of *D* (nm) and *h* (nm):

$$
\lambda\_0(LSPR) \approx \frac{160 + D + 3.86h}{0.319 + 0.0084h} \tag{7}
$$

A comparison of the simple model (Eq. 7) and experimental data in [21] shows a good agreement. Equation 7 seems to be a good approximation of the variation of the LSPR as a function of the geometrical parameters *D* and *h*, for a grating of rough gold cylinders with adhesion layer (*e* = 2 nm) and *P* = 200 nm (Fig. 1). The specific case of homothetic cylinders with *h* = *D*, removes a degree of freedom and leads to a simpler formula:

$$
\lambda\_0(LSPR, h = D) = \frac{160 + 4.86D}{0.319 + 0.0084D} \tag{8}
$$

This formula is strictly valid in the interval *h* = *D* ranging from 50 to 70 nm, by cons it should still be checked for greater heights.

Asymptotic form of *λ*0(*LSPR*) could be of interest especially for specific aspect ratio of the cylinder. For example, for nanodiscs (*h* << *D*), the position of the LSPR can be deduced from the series of Eq. 7.

$$
\lambda\_0(LSPR, h < < D) = (501 + 3.1D) - (0.0825D + 1.1)h + o[(h/D)^2],\tag{9}
$$

where *o*[(*h*/*D*)2] represents omitted terms of order higher than 2 in the series.

This equation confirms that the correction induced by the height assuming a small aspect ratio *h*/*D* is a blueshift of the LSPR, whatever are the diameters *D*. These formula can be used to select the best parameters for a given LSPR position. Another interest of these simple laws is the fact that propagation of uncertainties of the fabrication process can be evaluated directly, and therefore, the sensitivity analysis of LSPR position on the geometrical parameters *h* and *D* can be deduced, at least within the investigated domain of geometrical parameters and wavelengths. This sensitivity analysis is conducted in what follows.

#### **4.2. Sensitivity of LSPR to uncertainties on size parameters**

The propagation of uncertainties and the sensitivity analysis of a process or a physical phenomenon helps to improve the fabrication process of any device, by exhibiting the parameters that should be controlled first. Indeed the identification of critical parameters which uncertainty should be reduced is relevant, for a challenging improvement of technology.

First, the propagation of experimental uncertainties of fabrication can be deduced from the derivative of Eq. 7. The uncertainty on *λ*0(*LSPR*) is deduced form the uncertainties on *D* (*u*(*D*)) and *h* (*u*(*h*)) [44, 5.1.2]:

$$u(\lambda\_0(LSPR)) = \sqrt{\left(\frac{\partial \lambda\_0(LSPR)}{\partial D}\right)^2 u^2(D) + \left(\frac{\partial \lambda\_0(LSPR)}{\partial h}\right)^2 u^2(h)}\tag{10}$$

where the partial derivatives are called sensitivity coefficients. Using Eq. 7 gives:

<sup>324</sup> State of the Art in Biosensors - General Aspects Nanostructured Biosensors: Influence of Adhesion Layer, Roughness and Size on the LSPR: A Parametric Study 15 Nanostructured Biosensors: Influence of Adhesion Layer, Roughness and Size on the LSPR: A Parametric Study http://dx.doi.org/10.5772/52906 325

14 State of the Art in Biosensors / Book 1

should still be checked for greater heights.

from the series of Eq. 7.

technology.

(*u*(*D*)) and *h* (*u*(*h*)) [44, 5.1.2]:

*u*(*λ*0(*LSPR*)) =

A comparison of the simple model (Eq. 7) and experimental data in [21] shows a good agreement. Equation 7 seems to be a good approximation of the variation of the LSPR as a function of the geometrical parameters *D* and *h*, for a grating of rough gold cylinders with adhesion layer (*e* = 2 nm) and *P* = 200 nm (Fig. 1). The specific case of homothetic cylinders

*<sup>λ</sup>*0(*LSPR*, *<sup>h</sup>* <sup>=</sup> *<sup>D</sup>*) = <sup>160</sup> <sup>+</sup> 4.86*<sup>D</sup>*

This formula is strictly valid in the interval *h* = *D* ranging from 50 to 70 nm, by cons it

Asymptotic form of *λ*0(*LSPR*) could be of interest especially for specific aspect ratio of the cylinder. For example, for nanodiscs (*h* << *D*), the position of the LSPR can be deduced

This equation confirms that the correction induced by the height assuming a small aspect ratio *h*/*D* is a blueshift of the LSPR, whatever are the diameters *D*. These formula can be used to select the best parameters for a given LSPR position. Another interest of these simple laws is the fact that propagation of uncertainties of the fabrication process can be evaluated directly, and therefore, the sensitivity analysis of LSPR position on the geometrical parameters *h* and *D* can be deduced, at least within the investigated domain of geometrical

The propagation of uncertainties and the sensitivity analysis of a process or a physical phenomenon helps to improve the fabrication process of any device, by exhibiting the parameters that should be controlled first. Indeed the identification of critical parameters which uncertainty should be reduced is relevant, for a challenging improvement of

First, the propagation of experimental uncertainties of fabrication can be deduced from the derivative of Eq. 7. The uncertainty on *λ*0(*LSPR*) is deduced form the uncertainties on *D*

2

*u*2(*D*) +

 *∂λ*0(*LSPR*) *∂h*

2

*u*2(*h*), (10)

where *o*[(*h*/*D*)2] represents omitted terms of order higher than 2 in the series.

parameters and wavelengths. This sensitivity analysis is conducted in what follows.

**4.2. Sensitivity of LSPR to uncertainties on size parameters**

 *∂λ*0(*LSPR*) *∂D*

where the partial derivatives are called sensitivity coefficients. Using Eq. 7 gives:

*<sup>λ</sup>*0(*LSPR*, *<sup>h</sup>* << *<sup>D</sup>*)=(<sup>501</sup> + 3.1*D*) − (0.0825*<sup>D</sup>* + 1.1)*<sup>h</sup>* + *<sup>o</sup>*[(*h*/*D*)2], (9)

0.319 <sup>+</sup> 0.0084*<sup>D</sup>* (8)

with *h* = *D*, removes a degree of freedom and leads to a simpler formula:

$$u(\lambda\_0(LSPR)) = \sqrt{\left(\underbrace{\frac{1}{0.319 + 0.0084h}}\_{S\_D}\right)^2 u^2(D) + \left(\underbrace{\frac{0.11266 + 0.0084D}{(0.319 + 0.0084h)^2}}\_{S\_k}\right)^2 u^2(h)}\_{S\_k} \tag{11}$$

Knowing the experimental uncertainties on *h* and *D*, the uncertainty on the LSPR can be deduced. The sensitivity coefficients *Sh* and *SD* can be used to evaluate the effect of the experimental dispersion of values of *D* and *h* around a mean value. *SD* is almost independent of *h* and equal to 3. Regarding *Sh* and within the same approximation *Sh* ≈ 1.107 + 0.082*D*. Therefore, in the investigated domain of *D*, *Sh* ∈ [5.2; 21.6]. The uncertainty on *h* must be 7 times smaller than that on *D* to balance the contributions of each uncertainty to the overall uncertainty. This condition is becoming increasingly critical as the diameter *D* of cylinders is increased. Fortunately, the control on the thickness of gold deposition (height of cylinders) is about ±2 nm and that of diameters is about ±20 nm. In this case, the uncertainty on the LSPR position remains lower than 30 nm for all diameters if *h* > 40 nm (Fig.8).

**Figure 8.** Uncertainty on the position of the LSPR computed from the heuristic law (Eq. 11) deduced from the least square fit of the parametric study results. The parameters of the model are *P* = 200 nm with roughness (*rms* = 1.6 nm) and chromium adhesion layer (*e* = 2 − 6 nm).

Figure 9 gathers the experimental results retrieved from [23] with experimental uncertainties (gray domain and) the heuristic law deduced from Eq. 7 with uncertainties (Eq. 10). The good agreement between experimental results and the heuristic model with associated uncertainties can be observed. The heuristic model fits well the experimental data and the uncertainties deduced from Eq. 11 are coherent with the experimental ones.

In this section, we have obtained an heuristic law to describe the link between the position of the LSPR and the size parameters *D* and *h* of the cylinders. This law seems to be valid for an adhesion layer of chromium of thickness *e* = 2 − 6 nm. Moreover, the simplicity of this law helps to determine a first approximation of the size of the cylinder, as well as the effect of the propagation of experimental uncertainties on the position of the LSPR.

**Figure 9.** Experimental data retrieved from [23, Fig. 2] (*h* = 50 nm, *e* = 3 nm, *P* = 200 nm) (black) with uncertainties (gray). Heuristic model deduced from the numerical parametric study of gold nanocylinder grating (red curve) with *P* = 200 nm, *h* = 50 nm, RMS 1.6 nm and chromium adhesion layer (*e* = 4 nm).

#### **5. Conclusion**

A model based on the Discrete Dipole Approximation (DDA) is used for a parametric study of the biosensors made of a grating of nanocylinders. The Localized Surface Plasmon Resonance (LSPR) is modeled by the position of the maximum over the spectrum of the extinction cross-section. The investigated parameters are the height *h*, the diameter *D*, the thickness *e* of the chromium adhesion layer, and the roughness of the nanocylinders. Thin adhesion layers slightly modify the position of the LSPR but degrade significantly its quality: the sensitivity of the biosensor can be highly decreased. Roughness induces a redshift of the LSPR but less alters the quality of the resonance than the adhesion layer. The relative influence of *h* and *D* is more complex. Actually, if *D* increases, a redshift is always observed. The same effect occurs if *h* decreases. Consequently the fine tuning of the LSPR can be achieved by varying *D* or *h*, assuming a fixed distance between cylinders (*P*). This property helps to deduce heuristic laws for the LSPR position and the propagation of uncertainties. The basic law, deduced from a least square fit, gives simply an approximation of the position of the LSPR as a function of *D* and *h*. The agreement with experimental results [21, 23] is satisfactory, falling within the experimental uncertainties of fabrication. Therefore the heuristic law of behavior of the LSPR position can be used with confidence (for *P* = 200 nm). Varying the period *P* + *D* of the grating of nanocylinders could be of interest. The characterization of the LSPR being clarified, the influence of the functionalization layer will be the object of future studies.

#### **Acknowledgment**

This work was supported by the "Conseil Régional de Champagne Ardennes", the "Conseil Général de l'Aube" and the *Nanoantenna* European Project (FP7 Health-F5-2009-241818).

## **Author details**

Sameh Kessentini and Dominique Barchiesi

Automatic Mesh Generation and Advanced Methods (GAMMA3 - UTT/INRIA) - Charles Delaunay Institute - University of Technology of Troyes - Troyes, France

## **References**

16 State of the Art in Biosensors / Book 1

**5. Conclusion**

*h* = 50 nm, RMS 1.6 nm and chromium adhesion layer (*e* = 4 nm).

will be the object of future studies.

Sameh Kessentini and Dominique Barchiesi

**Acknowledgment**

**Author details**

**Figure 9.** Experimental data retrieved from [23, Fig. 2] (*h* = 50 nm, *e* = 3 nm, *P* = 200 nm) (black) with uncertainties (gray). Heuristic model deduced from the numerical parametric study of gold nanocylinder grating (red curve) with *P* = 200 nm,

A model based on the Discrete Dipole Approximation (DDA) is used for a parametric study of the biosensors made of a grating of nanocylinders. The Localized Surface Plasmon Resonance (LSPR) is modeled by the position of the maximum over the spectrum of the extinction cross-section. The investigated parameters are the height *h*, the diameter *D*, the thickness *e* of the chromium adhesion layer, and the roughness of the nanocylinders. Thin adhesion layers slightly modify the position of the LSPR but degrade significantly its quality: the sensitivity of the biosensor can be highly decreased. Roughness induces a redshift of the LSPR but less alters the quality of the resonance than the adhesion layer. The relative influence of *h* and *D* is more complex. Actually, if *D* increases, a redshift is always observed. The same effect occurs if *h* decreases. Consequently the fine tuning of the LSPR can be achieved by varying *D* or *h*, assuming a fixed distance between cylinders (*P*). This property helps to deduce heuristic laws for the LSPR position and the propagation of uncertainties. The basic law, deduced from a least square fit, gives simply an approximation of the position of the LSPR as a function of *D* and *h*. The agreement with experimental results [21, 23] is satisfactory, falling within the experimental uncertainties of fabrication. Therefore the heuristic law of behavior of the LSPR position can be used with confidence (for *P* = 200 nm). Varying the period *P* + *D* of the grating of nanocylinders could be of interest. The characterization of the LSPR being clarified, the influence of the functionalization layer

This work was supported by the "Conseil Régional de Champagne Ardennes", the "Conseil Général de l'Aube" and the *Nanoantenna* European Project (FP7 Health-F5-2009-241818).

Automatic Mesh Generation and Advanced Methods (GAMMA3 - UTT/INRIA) - Charles

Delaunay Institute - University of Technology of Troyes - Troyes, France


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