Study of Signal Transmission through different Array of Cu , Au and Ag Nanospheres

Twenty first century is the age of Information Technology and we always look for faster information transport and processing capabilities. Data can be moved by transistors and optical fibers. Transistors carry small amount of data and are small in size while optical fibers can carry huge amount of data but are much bigger in size. Metal nanostructures may possess right combination of electric and optical properties to tackle the issues outlined above and realized the dream of significantly faster processing speeds. In this work, dispersion relation in Plasmon modes in linear chain and alternated chain of nanoparticles like silvergold, silver-copper and gold-copper were studied. Expressions of angular frequency of Plasmon modes, group-velocity and extinction-coefficient were derived by solving equation of motion of Plasmon oscillation on the single kind of nanoparticle and that on different alternated chain of nanoparticles. Expressions of angular frequency of Plasmon modes, group velocity and extinction coefficient were tested for specific linear chain of nanoparticles. Extinction coefficients for the linear chain of Copper, gold and silver nanoparticles were found to be less and group velocities higher than that of alternated silver-gold and alternated gold-copper at their respective resonance frequencies.


INTRODUCTION
Quantum of collective longitudinal excitation of conduction electron on the surface is called Surface Plasmon (Kittle 2004).This excitation of conduction electron on the metal surface can be achieved using beam of accelerated electron and electromagnetic (EM) radiation as well.In 1957, Refuse Ritchie (Ritchie 1957) published a finding on electron energy losses in thin films, in which it was shown that plasmon modes can exist near the surface of metals.It was the first theoretical description of surface plasmons.In 1970 Uwe Keribig and Peter Zacharias (Kreibig & Zacharias 1970), performed a study in which they compared the electronic and optical responses of gold and silver nanoparticles.For the first time they described the optical properties of metal nanoparticles in terms of surface plasmon.In case of noble metal nanospheres, this exciting EM radiation lies in the range of the visible and nearinfrared energy range.Excited plasmon by the interaction of light radiation with electron cloud is called Plasmon Polariton (Barnes et al. 2003, Brongersma et al. 2007).At first glance, the use of metallic structures to transmit light signals seems impractical, because metals are known for high optical loss.But using simulation, it has been shown that electromagnetic radiation can be transmitted through the array of closely spaced metallic nanospheres (Quinten et al. 1998& Atwater 2007).Using the Silver nanospheres chain below the diffraction limit of visible range light, possibility of transportation and extinction of signal had been studied by the Brongersma et al. 2000(Brongersma et al. 2000) using point dipole approximation.They showed that signal can be transmitted using such an array of metallic nanospheres even at the speed of 0.1c.At the early stage, information transporting devices used to be bulky and slow in performance.Discovery of semiconductor followed by the transistor greatly increased speed of propagation of signal and hence tuned new era in information transportation technology.In this process not only speed was greatly increased, as illustrated in Fig. 1, but also there was reduction in size of device using advance nanotechnology.These semiconductor and transistor based devices let us to design circuit that can transport signal in terms of electric current.
Nano-scale study of matter has allowed us to design transistor even within 50nm (Ozbay 2006).But their efficiency is bounded by the velocity of electron flow through the wire.
Optical signal carrying devices, like optical fiber can carry signal with efficiency > 1000 times (Ozbay 2006), is desirable to find out structures that can interconnect optical signal carrying device and electric signal carrying bulky device (example: optical fibers that are bulky in size compared to that of present very tiny nanoscale level chips and circuit elements like transistor, capacitor, etc.).Metallic nanospheres can interact with electromagnetic radiation and excites (LSPP) on it that oscillates at the frequency many times greater that of incident radiation (Barnes et al. 2003).This compresses signal many times to very short wavelength range so that signal can pass through structure below its diffraction limit.Thus, the scarcity of efficient and fast interconnector can be fulfilled by the nanospheres array of metal, as nanospheres array of metal can provide guide to light signal by the mean of electron cloud oscillation on its surface, and hence allows signal to propagate through it.This permits us to design whole ultrafast signal processing circuit within a nano-scale range with ultrafast speed (Atwater 2007).

MATERIALS AND METHODS
This work tried to describe attenuation coefficients and signal propagation velocity, for signal that lies within visible and red infrared region, through array of linear and alternated nanospheres of Copper (Cu), Gold (Au) and Silver (Ag) based on point dipole approximation.In this analytical method of investigation, we have used classical electrodynamics' concept that can deal with oscillating dipole and their interactions.
Under the incident monochromatic electromagnetic (EM) radiation of wavelength, , greater than that of radius, , and inter-particle spacing between neighboring nanospheres, , there are homogenous polarization of nanospheres in the array and get polarized long the direction of incident electric field giving Longitudinal mode (L-mode) and Transverse mode (T-mode) of polarization depending upon the polarization of nanopsheres along the axis and perpendicular to the axis of nanospheres array respectively.

Electric field by an individual oscillating dipole
, due to types of material located at position in mode of polarization, at the neighboring nanospheres of types of material located at and position, depends upon the polarization dependent constant , is given by an equation (Brongersma et al. 2000), Electric field is dependent of and hence there is significance of near field rather than far field in coupling phenomena.Here we have used for dielectric constant of free space, and for L-mode and T-mode respectively such that and considering for Cu, Au and Ag respectively.Dipole on Localized Surface Plasmon (LSPP) at type of nanospherematerial oscillates with frequency , can be obtained in terms of bulk plasma frequency of types of material given by Drude model of metal.Due to oscillation of dipole with Eigen frequency of oscillation, , there is radiation loss and its affect can be expressed in term of relaxation frequency, , with (Jackson 2013) such that and are representing electric charge, optical effective mass of an electron, and velocity of light respectively.Oscillating LSPP also suffers damping due to non radiative loss like electron hole formation, hot electron formation, and phonon excitation, that depends on the Fermi velocity, , bulk mean free path, , and radius of nanosphere, , and hence give heating effect.This gives relaxation frequency for type of material nanosphere, , given by the relation (Kreibig & Genzel 1985), (2) There is coupling between nearby nanospheres through the electric field interaction between oscillating dipoles, and hence give rise coupling frequency of types of nanospheres, , depends upon the dipole charge, , as well as center to center spacing between nanospheres, , and optical effective mass of electron, , as given by equation (Brongersma et al. 2000) (Johnson & Christy 1972), where is rest mass of electron.On using bulk electron densities for Cu, Au and Ag as , and respectively (Kittle 2004), we got their respective coupling frequencies, , and , and their individual nanosphere oscillation frequencies, (Chen et al. 2011), and .Here we see coupling frequencies and resonance oscillation frequencies of individual dipole for our considered systems are of same order and hence there are possibilities of strong coupling interactions.Fermi velocities of electron on Cu, Au and Ag bulk materials, , and (Kittle 2004) together with their respective bulk mean free paths, (Sattler 2016), (Waser 2012) and (Brongersma et al. 2000) respectively.This results, their correspondence relaxation frequencies are rad/s, and respectively.Characteristic time period of oscillation of electrons in Cu, Au and Ag, , and respectively, give their respective relaxation time frequencies due to radiation loss, , and respectively.From here we get , and can conclude that effect on damping of nanospheres due to radiation loss is negligible than that of non-radiative loss.

(i) Dispersion Curve:
On the basis of equation ( 6

Fig. 3. Dispersion curve for alternated nanospheres array of (a) Cu and Au nanospheres, denoted by red color and (b) Au and Ag nanospheres, denoted by cyan colored lines. Here, dotted lines are representing their corresponding T-modes and solid lines are representing corresponding L-modes.
In Fig. 3, mean value of about which its value oscillates with variation of wave vector is larger for array of alternated Au and Ag nanospheres than that of alternated Cu and Au nanospheres array.Thus array of alternated of Au and Ag nanospheres array can carry signal of slightly upper energy range than that can be carried by alternated Cu and Au nanospheres array.Dislike of array of nanospheres of only one type of metallic nanospheres, in alternated array of nanospheres of different material, there is variation of at T-mode and L-mode in same direction with the change in wave vector .It helps us to generalize that, group velocity variation at T-mode and L-mode occurs along the same direction with the change in signal frequency .

(ii) Group Velocity:
We have presented group velocity at L-mode only as there is T-mode group velocity is half of the L-mode.

Fig. 4. Group velocity through the array of (i) (a) Cu, denoted by solid line, (b) Au, denoted by dotted line, (c) Ag, denoted by half dotted line and (ii) (a) alternated Cu and Au, denoted by solid line and (b) alternated Au and Ag, denoted by dotted line, nanospheres.
Fig. 4 shows that there is gradual increase in group velocity and reaches to peak value, corresponds to the central value of , resonance oscillation frequency in case of single material nanospheres chain, about which there is variation in value of in dispersion curve, and then again reaches to minimum zero value.There is existence of negative group velocity notifies that signal propagation and phase velocities are exiting in opposite direction.Velocity grows from negative to positive value as there is change in from minimum to mean value of it (resonance oscillation frequency in array of same types of nanospheres).This plot also shows that they have nearly same velocity of signal propagation through array of Cu, array of Au and array of Ag nanopsheres as their respective group velocities are , and respectively in L-mode polarization.We also got their respective values in T-mode; , and respectively in close agreement with previous carried out work (Brongersma et al. 2000).Furthermore, figure (4) shows that there is decrease in group velocity on using alternated nanospheres array of two different material as at L-modes in array of Cu-Au is and in array of Au-Ag is .Frequency within which signal can propagate through the array of alternated nanospheres of Cu and Au starts earlier than that of alternated array of Au and Ag nanosphers.Using the array of alternated nanospheres of Au and Ag, it is possible to transmit slightly larger volume of data as its frequency range, in which signal can propagate, is slightly larger than array of alternated Cu and Au nanospheres.Minimum attenuation coefficient for array of alternated Cu and Au nanoapheres is and that of alternated nanospheres of Au and Ag is .These values corresponds to attenuation , and .From equation ( 8), it is clearly seen that for small group velocity, there is very large attenuation.On the other hand we have discussed that at L-mode, signal can pass almost two times faster than at T-mode and hence give rise attenuation nearly twice of corresponding Lmode.So it may not be desirable to discuss Tmode than L-mode.

CONCLUSION
Among the arrays of nanospheres of Cu, Au and Ag, we got that group velocities is highest in the array of Ag nanospheres, , and lowest in the array of nanospheres of Cu, . Their corresponding attenuation coefficients are smallest, and largest among them.Their band width in array of Ag nanospheres is larger, although by very small value, and hence it is better among those arrays of nanospheres.Among the array of alternated nanaopshers of Cu and Au, and alternated nanosphers array of Au and Ag, large bandwidth, small attenuation, and higher group velocity exist for array of alternated nanospheres of Au and Ag but not on other.Thus among these two alternated nanospheres array, Au and Ag combination is best.In comparison of alternated and non-alternated nanosphers array, array of Ag nanospheres array is best.Coupling between two different metallic nanospheres is found to be weaker than that exist between same types of metallic nanospheres.Further study can be carried out in the system of alternated nanosphers hosted in some other metallic substrate can help to increase coupling phenomena (Chen et al. 2011).To obtain relatively higher speed of signal propagation and lower attenuation further research can be carried out by changing the shape of nanostructures used in constructing array (Maier et al. 2002) and host material.

Fig. 1 .
Fig. 1.Progress in signal transporting device speed and dimension.
), dispersion equation for single type of nanospheres array of Cu, Au and Ag is illustrated in Fig. 2.Here we see that, frequency of signal, propagating through the array, oscillates about certain central value.Allowed frequency region for Cu nanospheres array is slightly lower than that of Au, and Ag has upper frequency region.Band width for the array of Cu, array of Au and array of Ag nanospheres shows almost same magnitudes.Thus array of individual Cu, Au and Ag nanospheres can carry nearly same volume of data although their efficiencies may be different.Dispersion curves for T-mode, denoted by dotted lines, has almost half band width of corresponding L-mode with same period of variation with wave vector .This signifies, Tmode excitation of LSPP on array of nanospheres cannot propagate large amount of data as much as that can be done by corresponding L-mode.This phenomenon also appears on the array of alternated Cu and Au nanospheres, and array of alternated Au and Ag nanospheres as shown in Fig. 3.This phenomenon together with equation (7) of velocity shows that signal can propagate in L-mode almost two times faster than that of corresponding T-mode.Dispersion curves for array of alternated nanosperes made by Cu and Au, and that for array of alternated nanospheres made by Au and Ag is shown in Fig. 3.

(
iii) Attenuation Coefficient: Using the equation (8), we can obtain attenuation coefficient for array of single types of nanospheres material as well as two types of nanospheres material.Attenuations are large at the extreme value of the frequencies and almost constant within these values.This phenomenon also appears in the case of alternated nanospheres array as given in Fig. 5.There is minimum attenuation at the resonance oscillation frequency of individual nanospheres at which there exists maximum group velocity in each chain.Minimum attenuation coefficient for array of nanospheres of Cu is largest, , and of Ag is smallest, , among arrays of Cu, array of Au and array of Ag nanospheres.For the array of Au nanospheres, it is at L-mode of polarization.These values, attenuation coefficients, get doubled in T-mode.

Fig. 5 .
Fig. 5. Attenuation coefficient for the nanospheres array of (i) (a) Cu, denoted by solid line, (b) Au, denoted by dotted line , (c) Ag, denoted by half dotted line and (ii) (a) alternated Cu and Au, denoted by solid line, (b) alternatedAu and Ag, denoted by dotted line.

,
With above consideration, equation of oscillation of nanospheres array, made by two different material of types and , is given by the relation,