Values of the model parameters.

## Abstract

In this work, we model the dielectric functions of gold (Au) and silver (Ag) which are typically used in photonics and plasmonics. The modeling has been performed on Au and Ag in bulk and in nanometric states. The dielectric function is presented as a complex number with a real part and an imaginary part. First, we will model the experimental measurements of the dielectric constant as a function of the pulsation ω by appropriate mathematical functions in an explicit way. In the second part we will highlight the contributions to the dielectric constant value due to intraband and interband electronic transitions. In the last part of this work we model the dielectric constant of these metals in the nanometric state using several complex theoretical models such as the Drude Lorentz theory, the Drude two-point critical model, and the Drude three-point critical model. We shall comment on which model fits the experimental dielectric function best over a range of pulsation.

### Keywords

- Modeling bulk and nanometric dielectric function
- noble metals Au and Ag
- interband transitions
- intraband transitions
- UV and IR pulsations

## 1. Introduction

All the intrinsic effects corresponding to the process of light-matter interaction are contained in the dielectric function

where

Several important physical quantities can be deduced from the complex refractive index

## 2. Modeling of the bulk experimental dielectric function

Here we try to model the experimental dielectric function * et al.*[10], Leveque

*[11], and Thèye*et al.

*[12] respectively. These measurements are summarized by Rakiç*et al.

*[13] in Figure 1 (for Ag), and in Figure 2 (for Au).*et al.

### 2.1 Modeling the experimental dielectric function of bulk Ag

In this part, we will model the real and imaginary parts of the experimental dielectric function of Ag (Figure 1). For this we will divide the values of the pulsation ω into several intervals in order to allow us to determine the best fit to suitable mathematical functions over a certain interval. All the results will be presented in Tables 1–8.

Parameter | R-Square (COD) | |||||
---|---|---|---|---|---|---|

Value | 10499.713 | −1300.876 | 3.813 × 10^{14} | −9163.320 | 1.217 × 10^{14} | 0.9999 |

Parameter | R-Square (COD) | |||||||
---|---|---|---|---|---|---|---|---|

Value | 0.305 | 83.392 | 1.091 × 10^{15} | 86.023 | 1.091 × 10^{15} | 86.639 | 1.091 × 10^{15} | 0,99101 |

Parameter | R-Square (COD) | |||||
---|---|---|---|---|---|---|

Value | 5.281 | −0.006 | −9.588 × 10^{14} | −0.006 | −9.576 × 10^{14} | 0.99671 |

Parameter | R-Square (COD) | ||||
---|---|---|---|---|---|

Value | 0.868 | 0.735 | −9.816 × 10^{14} | 3.483 × 10^{14} | 0.96592 |

Parameter | R-Square (COD) | |||||
---|---|---|---|---|---|---|

Value | 2.539 | 7.140 × 10^{15} | −2.399 | 6.978 × 10^{14} | 3.783 × 10^{24} | 0.99613 |

Parameter | R-Square (COD) | ||||
---|---|---|---|---|---|

Value | −23218.730 | 585.784 | 2.267 × 10^{14} | 6.910 × 10^{13} | 0.99998 |

306.100 | 1.844 × 10^{14} | 23235.380 | 2.514 × 10^{18} |

Parameter | R-Square (COD) | ||||
---|---|---|---|---|---|

Value | 7.076 | −6.327 | 1.38101 × 10^{15} | 4.92416 × 10^{14} | 0.9983 |

Parameter | ^{−1} | |||||
---|---|---|---|---|---|---|

Value | 1.278 | 1.633 | −8.811 | 9.897 × 10^{15} | 1.262 × 10–^{16} | 6.159 × 10^{15} |

^{−1} | R-Square (COD) | |||||

3.945 × 10–^{15} | 0.99733 |

#### 2.1.1 Modeling the real part of the bulk dielectric function of Silver ε R − Ag exp ω

For

For

For

For

For

#### 2.1.2 Modeling the imaginary part of the bulk dielectric function of Silver ε I − Ag exp ω

For

For

For

### 2.2 Modeling the experimental dielectric function of bulk Au

In this section, we will model the real and imaginary parts of the experimental dielectric function of bulk Au (Figure 2). For this we will proceed in the same way as for bulk Ag by dividing the values of the pulsation ω into various intervals. All the results are listed in Tables 9–15.

Parameter | ^{−1} | ^{−2} | ^{−3} | ^{−4} | R-Square (COD) | |
---|---|---|---|---|---|---|

Value | 4835.830 | −8.633 × 10^{−12} | −6.930 × 10^{−26} | 3.779 × 10^{−40} | −8.746 × 10^{−55} | |

^{−5} | ^{−6} | ^{−7} | ^{−8} | ^{−9} | 0.9999 | |

1.170 × 10^{−69} | −9.674 × 10^{−85} | 4.893 × 10^{−100} | −1.392 × 10^{−115} | 1.710 × 10^{−131} |

Parameter | ^{−1} | ^{−1} | R-Square (COD) | |||||
---|---|---|---|---|---|---|---|---|

Value | 7.786 | 138.106 | 0.896 | 1.495 × 10^{15} | −1.388 × 10^{−15} | 2.641 × 10^{15} | −3.366 × 10^{−15} | 0.9996 |

Parameter | ^{−1} | ^{−2} | ^{−3} | ^{−4} | ^{−5} | R-Square (COD) | |
---|---|---|---|---|---|---|---|

Value | −50227.482 | 6.432 × 10^{−11} | −3.281 × 10^{−26} | 8.338 × 10^{−42} | −1.056 × 10^{−57} | 5.337 × 10^{−74} | 0.9997 |

Parameter | ^{−1} | ^{−2} | ^{−3} | ^{−4} | R-Square (COD) | |
---|---|---|---|---|---|---|

Value | 271340.234 | −4.224 × 10^{−10} | 2.912 × 10^{−25} | −1.167 × 10^{−40} | 2.995 × 10^{−56} | |

^{−5} | ^{−6} | ^{−7} | ^{−8} | ^{−9} | 0.99872 | |

−5.112 × 10^{−72} | 5.801 × 10^{−88} | −4.220 × 10^{−104} | 1.786 × 10^{−120} | −3.351 × 10^{−137} |

Parameter | ^{−1} | ^{−2} | ^{−3} | ^{−4} | R-Square (COD) | |
---|---|---|---|---|---|---|

Value | 9873.897 | −9.845 × 10^{−11} | 4.479 × 10^{−25} | −1.188 × 10^{−39} | 2.002 × 10^{−54} | |

^{−5} | ^{−6} | ^{−7} | ^{−8} | ^{−9} | 0.9999 | |

−2.212 × 10^{−69} | 1.598 × 10^{−84} | −7.276 × 10^{−100} | 1.895 × 10^{−115} | −2.152 × 10^{−131} |

Parameter | ^{−1} | ^{−2} | ^{−3} | ^{−4} | ^{−5} | R-Square (COD) | |
---|---|---|---|---|---|---|---|

Value | 115.516 | −1.891 × 10^{−13} | 1.299 × 10^{−28} | −4.538 × 10^{−44} | 7.892 × 10^{−60} | −5.361 × 10^{−76} | 0.9973 |

Parameter | ^{−1} | ^{−2} | ^{−3} | ^{−4} | R-Square (COD) | |
---|---|---|---|---|---|---|

Value | −53432.059 | 1.046 × 10^{−13} | −8.722 × 10^{−26} | 4.106 × 10^{−41} | −1.210 × 10^{−56} | |

^{−5} | ^{−6} | ^{−7} | ^{−8} | ^{−9} | 0.9974 | |

2.324 × 10^{−72} | −2.918 × 10^{−88} | 2.316 × 10^{−104} | −1.056 × 10^{−120} | 2.111 × 10^{−137} |

#### 2.2.1 Modeling the real part ofthe bulk dielectric function of Gold ε R − Au exp ω

For

For

For

For

#### 2.2.2 Modeling the imaginary part ofthe bulk dielectric function of Gold ε I − Au exp ω

For

For

For

All the results of these different models for both bulk Ag and bulk Au are plotted in the following Figures 3 and 4.

In both cases, the experimental results can be well fitted by the models of experimental dielectric function in both its real and imaginary parts with mathematical functions with high accuracy.

## 3. Highlighting the contribution of interband and intraband transitions in the expression of dielectric function

In metals, there aretwo types of contribution in the dielectric function, namely contribution of interband transitions denoted

The first term corresponds to the intraband component of the dielectric function. It is referred to the optical transitions of a free electron from the conduction band to a higher energy level of the same band. The second term corresponds to the interband component of the dielectric constant. It is referred to optical transitions between the valence bands (mainly d band and s-p conduction bands). Due to Pauli’s exclusion principle, an electron from a valence band can only be excited to the conduction band. There is therefore an energy threshold

The intraband part

where

We note that

where

In the Drude model, there appears a pulsation called the plasmon frequency of a bulk metal given by

Where

The electronic structure of bulk noble metals such as Ag and Au, the respective values of the conduction electron density

### 3.1 Contribution of intraband transitions to dielectric function

The intraband dielectric function described by the Drude model [14] as denoted

The real and imaginary parts of the relative dielectric function (intraband) are written as follows

Usually, for noble metals

The following Table 17 shows the values of plasma frequencies and the collision rate of noble metals (Gold, Silver):

Metal | ||||
---|---|---|---|---|

Ag | 5.86 | 0.96 | 1.39 | 55 |

Au | 5.90 | 0.99 | 1.40 | 42 |

Metal | ^{2} | ^{−1}) |
---|---|---|

Ag | ||

Au |

The results of the calculations of the contribution of intraband effects to dielectric function are represented in their real and imaginary parts for bulk Ag (Figure 5) and for bulk Au (Figure 6).

For the noble metals (Gold, Silver), we observed that the real and imaginary parts decrease withincreasing pulsation. The further away from the pulsations corresponding to IR radiation and the closer we get to the pulsations corresponding to UV radiation, these values decrease.

### 3.2 Contribution of interband transitions to dielectric function

The interband dielectric function denoted

The real and imaginary parts of the interband dielectric function are written respectively as follows:

where

Here

The results of the calculations of the contribution of interband transitions to real and imaginary parts of dielectric function are represented respectively for bulk Ag (Figure 7) and for bulk Au (Figure 8).

As shown in Figure 7, the real part of the contribution of interband effects to the dielectric function of bulk Ag decreases with increasing pulsation in the IR radiation domain then is still almost constant with small variations from

We note that for Gold, the real and imaginary parts of the contribution of interband transitions decrease by increasing the values of the pulses in the IR domain to the value

In Figures 9 and 10 (for Ag) and Figures 11 and 12 (for Au), we have presented experimental values, the contributions of intraband and interband transitions to the real and imaginary parts of the dielectric function.

Concerning the real part, we note that, the real part due to the intraband transitions noted

Concerning the imaginary part, we note that the imaginary part due to the interband transitions noted

For Gold, we note that, the real part due to the intraband transitions noted

Concerning the imaginary part, as for Silver, we see that the imaginary part due to the interband transitions noted

## 4. Modeling the dielectric functions of nanometri cAg and nanometric Au

In this section, we study the dielectric functions of nanometric Ag and nanometric Au. They are composed of a few tens to several thousand atoms. Their very small characteristic dimensions, in the nanometer range (i.e. well under optical wavelengths), give rise to extraordinary electronic and optical properties that cannot be observed in bulk materials. These properties are clearly influenced by the size, form of the nanoparticle and the nature of the host environment. We consider the measured values of dielectric function used in the previous paragraphs, and try to model those using theoretical models for nanometals such as the Drude Lorentz (DL) model, the Drude two-point critical model DCP and the Drude three-point critical model DCP3.

The optical properties of metallic nanoparticles are dominated by the collective oscillation of conduction electrons induced by interaction with electromagnetic radiation (IR, UV).

The collective excitation of nanoparticle conduction electrons gives them new optical properties; we consider the following two effects:

Plasmons guided along a metallic film of nanometric cross-section (1D confinement).

Surface plasmons located in a metallic particle of nanometric size (0D confinement).

### 4.1 Drude Lorentz (DL) model

For the study of resonant nanostructures, it is important to have a good description of the permittivity of the metal in a large frequency band. For this purpose, the validity band of the Drude Model is often extended by adding Lorentzian terms [16] depending in the following form

The dielectric function described by the Drude Lorentz model is written as follows

where:

The real part of the dielectric function according to the DL model

The imaginary part of the dielectric function according to the DL model:

The studies of Vial and Laroche [16] on the permittivity of Auand Ag metals used in their model with the parameters are listed in Table 18.

(rad/s) | γ (rad/s) | ||||
---|---|---|---|---|---|

Au | 6.2137 | 1.3323 × 10^{16} | 1.3235 × 10^{14} | 3.4620 | |

Ag | 1.7984 | 1.3359 × 10^{16} | 8.7167 × 10^{13} | 3.0079 | |

(rad/s) | (rad/s) | (rad/s) | (rad/s) | ||

Au | 4.7914 × 10^{15} | 2.1367 × 10^{15} | −3.4886 | 4.2111 × 10^{14} | 4.5572 × 10^{17} |

Ag | 8.1635 × 10^{15} | 4.3785 × 10^{17} | 2.3410 | 3.8316 × 10^{16} | 6.0574 × 10^{16} |

The results of modeling the experimental dielectric function in its real and imaginary parts using the DL model are shown in Figures 13 and 14 for Au(in Figures 15 and 16 for Ag).

### 4.2 Drude model with two critical points DCP

In order to describe the metal in the largest possible range of pulsations, another formula describing the two-point critical Drude model (DCP) [16] will appear in this paragraph.

The dielectric function of Au and Ag is can be expressed from as [17]:

The dielectric function described by the Drude two-critical-point model is written as follows

where

The real part of the dielectric function according to the DCP model:

The imaginary part of the dielectric function according to the DCP model:

The work of Alexandre Vial’s [17] on the permittivity of the noble metals (Gold, Silver) made model with the parameters are listed in Table 19.

(rad/s) | γ (rad/s) | (rad) | (rad/s) | |||
---|---|---|---|---|---|---|

Au | 1.1431 | 1.3202 × 10^{16} | 1.0805 × 10^{14} | 0.26698 | −1.2371 | 3.8711 × 10^{15} |

Ag | 15.833 | 1.3861 × 10^{16} | 4.5841 × 10^{13} | 1.0171 | −0.93935 | 6.6327 × 10^{15} |

(rad) | (rad/s) | (rad/s) | (rad/s) | |||

Au | 3.0834 | −1.0968 | 4.1684 × 10^{15} | 2.3555 × 10^{15} | 4.4642 × 10^{14} | |

Ag | 15.797 | 1.8087 | 9.2726 × 10^{17} | 2.3716 × 10^{17} | 1.6666 × 10^{15} |

The results of modeling the experimental dielectric function in its real and imaginary parts using the Drude two-point DCP critical point model are shown in Figures 13 and 14 for Au (in Figures 15 and 16 for Ag).

### 4.3 Drude model with three critical points DCP3

The DCP3 model describes the response of the dielectric function in a wider pulsation band, it should be noted that the DCP3 model gives a very good description of the dielectric function of noble metals; it is expressed by the relation [18]:

The dielectric function described by the Drude two-critical-point model is written as follows:

Where:

The real part of the dielectric function in the DCP3 model:

The imaginary part of the dielectric function in the DCP3 model:

The parameters of this model are given in Table 20.

(rad) | (rad/s) | (rad/s) | |||||
---|---|---|---|---|---|---|---|

Au | 1.1156 | 27.825 | 0.5548 | 2.8463 | 4.506 × 10^{16} | 5.09 × 10^{16} | 679.7606 |

Ag | 1.4783 | 8.7191 | 1.007 | −0.9621 | 6.617 × 10^{15} | 1.7415 × 10^{15} | 5377.4512 |

(rad) | (rad/s) | (rad/s) | (rad) | (rad/s) | (rad/s) | ||

Au | −0.0998 | 3.4587 × 10^{14} | 3.064E13 | 3.5244 | 4.6586 | 3.5832 × 10^{15} | 1.68784 × 10^{15} |

Ag | −0.0092 | 1.3545 × 10^{14} | 6.56505 × 10^{12} | 2.6077 | −2.8539 | 8.1007 × 10^{14} | 8.7193 × 10^{12} |

The results of modeling the real and imaginary parts of experimental dielectric function using the Drude three-point critical point model DCP3 are shown in Figures 13 and 14 for Au (in Figures 15 and 16 for Ag).

Concerning the real part of the dielectric function of nanometric Ag; the model that is very much in agreement with the experiment up to the value of the pulsation

For the imaginary part of nanometric Ag, we find that for pulsations located in the IR domain and less than

In the case of the real part of the dielectric function of nanometric Au, the DL, DCP, and DCP3 models are allin very good agreement with the experiment up to the value of the pulsation

## 5. Conclusion

In this work, we modeled the dielectric function of noble metals (silver and gold) in their bulk and nanometric states. Initially, we modeled the measured dielectric functions of these two metals using explicit mathematical functions and the results are in very good agreement with the experiment. Moreover, we have decomposed these measured values of the dielectric functions; in their real and imaginary parts; into several intervals according to the pulsations that sweep the domains corresponding to IR and UV radiation via the intermediate values. The obtained results are very conclusive, and depending on the pulsation domain studied, it is possible to use the corresponding mathematical function in simulations and calculations. Then, we highlighted the importance of the contributions of intraband and interband transitions in dielectric function for both Ag and Au. For Ag, we note that the imaginary part of the dielectric function due to interband transitions denoted

In the case of Au, we note that the real part of the dielectric function due for

In the last part of this paper, we have modeled the dielectric functions of Ag and Au, using theoretical models that deal with nanometric systems such as the Drude Lorenz model, the Drude two-point critical model, and the Drude three-point critical model.In the case of nanometic Ag, the real part of the dielectric function model agrees well with the experiment up to the value of the pulsation

For nanometric Au, concerning the real part of the dielectric function, the three models DL, DCP, and DCP3 are all in very good agreement with the experiment up to the value of the pulsation

## Acknowledgments

We are grateful to Professor Uwe Thumm who hosted us for three months in his James R. Macdonald laboratory at the Kansas State University in the USA, and who offered us an opportunity to collaborate on this subject, as part of the Fulbright Grant Merit Award.