WEBVTT Kind: captions Language: en 00:00:02.580 --> 00:00:07.020 Hello 00:00:31.320 --> 00:00:36.540 students, welcome to lecture 19 of the online course on Nanophotronics, 00:00:36.540 --> 00:00:44.100 Plasmonics and Metamaterials. Today's lecture will be on Localized Surface Plasmon Resonance or in 00:00:44.100 --> 00:00:51.360 short LSPR. So here is the lecture outline, we will first see what is localized surface plasmon 00:00:52.200 --> 00:01:00.900 and then we will see how to do the derivations of localized surface plasmon resonance conditions 00:01:00.900 --> 00:01:08.880 and we will treat using exact theory of LSPR that is Mie scattering theory or Mie theory. 00:01:09.420 --> 00:01:15.360 We will also look into the quasi-static approximation of LSPR. We will find out 00:01:15.360 --> 00:01:20.220 how to calculate the scattering and absorption cross section. We will also see how to handle 00:01:20.220 --> 00:01:26.520 the cases beyond quasi-static approximation and towards the end we will see the characteristics 00:01:26.520 --> 00:01:31.560 of void plasmon and metallic nano shells. So let us begin with localized surface plasmon. 00:01:31.560 --> 00:01:37.380 So we have seen in the previous lecture the surface plasmon are basically the propagating 00:01:37.380 --> 00:01:45.060 waves which we call them as SPPs, surface plasmon polaritons. These are propagating 00:01:45.060 --> 00:01:51.300 dispersive electromagnetic waves which are coupled to the electron plasma of a conductor 00:01:51.900 --> 00:01:58.980 and they are propagated metal dielectric interface. Now localized surface plasmon 00:01:58.980 --> 00:02:05.880 on the other hand are basically non-propagating that is why they are called localized and they are 00:02:05.880 --> 00:02:13.440 non-propagating excitations of the conduction electrons on metallic nanostructures which 00:02:13.440 --> 00:02:18.720 are coupled to the electromagnetic radiation. Now you can look into this particular figure here. 00:02:19.320 --> 00:02:26.520 It shows the illustration of a localized surface plasmon resonance. So this is a tiny metallic 00:02:26.520 --> 00:02:35.580 nanoparticle and with light falling on it with the electric field oscillating up and down the 00:02:35.580 --> 00:02:40.020 electron cloud is also getting so when the oscillate electric field is in this direction 00:02:40.920 --> 00:02:45.900 electron cloud is basically pushed downwards so that creates a kind of negative charge. 00:02:46.740 --> 00:02:53.520 All the electron clouds are pushed downwards so there is a kind of negative charge here you can 00:02:53.520 --> 00:03:00.180 say and there is lack of electrons on the top side so you can think of some positive charge 00:03:00.180 --> 00:03:06.540 formation there or there are some holes you can say. In that case you are able to 00:03:06.540 --> 00:03:12.960 see some charge separation that is positive and negative so this actually becomes like a dipole 00:03:12.960 --> 00:03:20.400 and as the electric field changes when the electric field is negative you will see the 00:03:20.400 --> 00:03:26.280 electron clouds are moved upwards in this case so you have the negative charges here and the 00:03:27.480 --> 00:03:32.040 absence of the electrons are felt here which are the positive charges and so 00:03:32.040 --> 00:03:36.060 the dipole also got reversed. So that is how with the electric 00:03:36.060 --> 00:03:42.840 field this metallic nanoparticle gets a induced dipole that also oscillates. Now with that what 00:03:42.840 --> 00:03:50.700 happens you might know this fact that oscillating dipole radiates. Now in this particular case this 00:03:50.700 --> 00:03:55.560 metallic nanoparticles they also behave like dipoles which are oscillating with 00:03:56.220 --> 00:04:04.800 incident electromagnetic field. Now you can analyze this problem using a scattering problem 00:04:04.800 --> 00:04:10.920 of a small sub wavelength size nanoparticle in an oscillating electromagnetic field. In this 00:04:10.920 --> 00:04:15.000 particular case it is important to remember that the curved surface of the nanoparticle 00:04:15.000 --> 00:04:24.720 exerts the restoring force on the driving or on the driven electrons so that they are pulled back 00:04:24.720 --> 00:04:30.900 and that is how this oscillation will start they kind of oval like a piece of jelly. 00:04:30.900 --> 00:04:38.400 So if you put a bulk of jelly on the table and try to poke it with a finger or a spoon you 00:04:38.400 --> 00:04:46.200 will see that the jelly is kind of wobbling. The similar kind of feature is also seen for surface 00:04:46.200 --> 00:04:53.520 electrons in this case. So this lead to a field amplification both inside and in the near field 00:04:53.520 --> 00:05:02.460 zone of this particular particle. So inside and near field of the particles will get some kind 00:05:02.460 --> 00:05:09.780 of amplification of the electromagnetic fields and that is what will give rise to resonance. 00:05:09.780 --> 00:05:16.320 So when this natural frequency of oscillation of the electrons matches the frequency of the 00:05:16.320 --> 00:05:20.460 incident field there is a resonance. That means those metallic nanoparticles 00:05:20.460 --> 00:05:27.420 are able to strongly absorb or scatter light much larger than the geometrical cross section and that 00:05:27.420 --> 00:05:33.360 is the phenomena of resonance and we call that resonance as localized surface Plasmon resonance 00:05:34.080 --> 00:05:42.060 or you can say Plasmon resonance. So here what is the good thing as compared to the case of SPPs 00:05:42.060 --> 00:05:48.300 that here you do not need to worry about the phase matching condition. So you can simply shine light 00:05:48.300 --> 00:05:57.540 and excite the Plasmons. So localized surface Plasmon can be excited by direct illumination. So 00:05:57.540 --> 00:06:02.940 the physics of localized surface Plasmon will be explored in this particular lecture by considering 00:06:02.940 --> 00:06:09.960 the interaction of the metallic nanoparticles with electromagnetic waves and we will see 00:06:09.960 --> 00:06:16.800 how do we get to the resonance condition. We will also see the damping process because 00:06:16.800 --> 00:06:22.440 whenever there is a resonator there are some damping associated that decides basically the 00:06:22.440 --> 00:06:28.980 Q factor of the resonator. So we will see how this damping process depends on the nanoparticle 00:06:28.980 --> 00:06:35.460 different sizes and shapes and how the interaction between the particles in an ensemble 00:06:35.460 --> 00:06:43.260 or in large assembly they actually affect this resonance. So along with that we will also see 00:06:44.280 --> 00:06:52.920 what are the other structures other than say solid nanoparticles that support this kind of resonance. 00:06:52.920 --> 00:06:59.760 So we will see that the dielectric inclusion in metallic bodies or you can say that a void in a 00:06:59.760 --> 00:07:05.100 metallic surface that can also support this kind of localized surface Plasmon resonance 00:07:05.100 --> 00:07:11.100 and also nano shells they can also support. So we will also look into this particular cases. 00:07:11.100 --> 00:07:20.160 Now when you look for surface Plasmon resonance the Plasmonic materials which are considered are 00:07:20.160 --> 00:07:25.620 basically gold and silver nanoparticles because they are also particularly interesting because 00:07:25.620 --> 00:07:32.520 their resonance falls in the visible range of the electromagnetic spectrum. So that 00:07:35.580 --> 00:07:43.980 you can see directly that the particles are able to transmit and reflect bright colors and they are 00:07:43.980 --> 00:07:49.860 basically coming from absorption and scattering which are enhanced because of the resonance. 00:07:50.820 --> 00:07:58.980 And this effect has been found several years back maybe hundreds of years back you can see 00:07:58.980 --> 00:08:07.080 like the Gothic stained glass in Notre Dame De Paris. So there all these beautiful and 00:08:07.080 --> 00:08:12.960 bright colors are basically coming from gold or silver nanoparticles embedded into the 00:08:12.960 --> 00:08:18.720 glass. So while making they used to make mix this metal to get this bright colors 00:08:18.720 --> 00:08:22.920 these are basically the colors coming out from the resonance of the nanoparticles. 00:08:23.580 --> 00:08:29.520 Basically the Lycurgus cup this cup looks different in color when it is illuminated 00:08:29.520 --> 00:08:37.740 from outside it shows it looks like a green cup ok. But when the light source is inside 00:08:37.740 --> 00:08:45.360 ok that is the case when you actually see the light that is what is not absorbed is basically 00:08:45.360 --> 00:08:51.420 coming towards you. So in that case from the white light source that is kept at the back 00:08:51.420 --> 00:08:59.700 of the cup if you remove the Plasmonic resonance that is at blue green you will see that the only 00:08:59.700 --> 00:09:05.040 red light is coming out towards you. So you will be seeing the cup as a red cup when the 00:09:05.040 --> 00:09:10.920 light source is behind. But when the light source is in the front you just see the scattering and 00:09:10.920 --> 00:09:16.320 the scattering resonance is at the blue green or simply green so the cup appears green in color. 00:09:16.920 --> 00:09:21.960 So it is the same cup but it looks different because of this Plasmonic properties. 00:09:22.560 --> 00:09:30.300 Now when we think of calculating a LSPR we need to first see is there any exact method of calculating 00:09:30.300 --> 00:09:41.880 the scattering and absorption from this particles nanoparticles. So the solution comes from the Mie 00:09:41.880 --> 00:09:50.460 theory. So in 1908 scientist ghost of Mie he was able to find an exact theory that can give 00:09:50.460 --> 00:09:57.300 explanation to the colors of different colloidal solutions of nanoparticles of different sizes. 00:09:58.680 --> 00:10:03.240 So there was an experiment where people made colloids of different size of nano 00:10:03.240 --> 00:10:06.840 gold and silver nanoparticles and all of them look different in color. 00:10:07.560 --> 00:10:11.400 So there has to be some relationship between the size of the particles 00:10:11.400 --> 00:10:16.740 with the wavelength and the color they strongly scatter or reflect. 00:10:17.700 --> 00:10:26.220 So that theory was known as Mie theory and how it works? This works from the concept that Maxwell's 00:10:26.220 --> 00:10:34.320 equations are linear. So you can think of a total field that is a kind of summation of 00:10:34.320 --> 00:10:41.100 the plane of excitation plus the outgoing wave that is the scattered wave plus the standing 00:10:41.100 --> 00:10:47.580 wave that is the wave inside that particular particle or void. So ghost of Mie actually 00:10:48.960 --> 00:10:55.260 did the solution for spherical particles. So on spherical coordinate systems he was able to solve 00:10:55.260 --> 00:11:02.160 the coefficients for each of the wave by matching the boundary conditions and he was able to find 00:11:02.160 --> 00:11:07.980 out the exact calculation of the scattering and absorption cross section by these particles. 00:11:08.700 --> 00:11:16.260 Now how it works? I will not go into the complete mathematical description of Mie theory. This can 00:11:16.260 --> 00:11:23.220 be found in this particular reference, the second reference as you see here, Bohren Hoffman book. So 00:11:23.220 --> 00:11:28.740 in this particular book, Absorption and Scattering of Light by Small Particles, you can actually 00:11:29.700 --> 00:11:35.940 see the complete derivation of Mie theory. So I am just showing the important formula here that 00:11:35.940 --> 00:11:42.660 will help you understand how this particular theory is derived and it is working. So time 00:11:42.660 --> 00:11:49.320 harmonic scalar wave equation, we can write it. So what is psi? Psi is basically the electric 00:11:49.320 --> 00:11:55.740 potential, it is function of r here. So this is the wave equation del square plus k square 00:11:57.000 --> 00:12:04.920 times psi r equals 0. Now electric and magnetic field if you express them as linear combinations 00:12:04.920 --> 00:12:17.220 of the vector harmonics M and N, you can write M equals grad cross r psi and N equals 1 over k 00:12:18.660 --> 00:12:28.260 curl of M. So these are kind of some relations with electric and magnetic field. So then if 00:12:28.260 --> 00:12:34.920 you try to use this particular equation for a spherical coordinates or a particle 00:12:35.640 --> 00:12:41.460 with spherical symmetry like a sphere or nanosphere, you can actually write 00:12:41.460 --> 00:12:50.340 down this in terms of spherical coordinates. So you have r, theta and phi coordinates. So now 00:12:50.340 --> 00:12:57.060 you are also able to write down your psi that is the electric potential in terms of r, theta, phi 00:12:57.060 --> 00:13:06.360 as three different variables. So you can separate these variables and say that this is basically R 00:13:06.360 --> 00:13:12.420 of r, theta of theta and phi of phi. So these are the three variables and you can actually write 00:13:12.420 --> 00:13:20.100 down the equation now in this particular form. So from here to here by assuming that psi r, 00:13:20.100 --> 00:13:25.620 theta, phi is basically this function. Now if you solve it only for the phi equation, 00:13:26.460 --> 00:13:33.180 you get phi equals e to the power plus minus i M small phi. You can solve it for the theta 00:13:33.180 --> 00:13:38.340 equation, you will come up the theta is basically associated Legendre polynomial. 00:13:38.880 --> 00:13:45.840 And when you solve it for the r function, that is the radius function, this is how it looks like and 00:13:45.840 --> 00:13:53.220 r turns out to be square root of 2 by pi Zl k r where Zl represents spherical Bessel function. 00:13:53.220 --> 00:14:02.160 And in this particular equation you will see that you will require jl that is basically spherical 00:14:02.160 --> 00:14:10.740 Bessel function of the first kind which is finite at r equals 0 and for rectangular waves you can 00:14:10.740 --> 00:14:19.020 use it for incident and internal cases. Also for the scattered waves you can think of spherical 00:14:19.020 --> 00:14:25.380 Hankel function which is given as small hl. So what is the specialty of this function? It 00:14:25.380 --> 00:14:33.660 can be written in the form of e to the power i k r over r where r tends to as r tends to infinity, 00:14:33.660 --> 00:14:42.720 okay. So you can understand that this scattered wave at infinity will die down, will go to 0. So 00:14:42.720 --> 00:14:51.360 with that you are able to express e and h of the incident field using the two vector harmonics M 00:14:51.360 --> 00:14:57.840 and N that you have seen here, okay. So you are writing them as a linear combination of these 00:14:57.840 --> 00:15:05.220 two vector harmonics. So h incident and e incident are the incident electric and magnetic fields. 00:15:05.220 --> 00:15:10.740 Now then you apply the conditions like for a plane wave with incident angle theta equals 0, 00:15:10.740 --> 00:15:17.580 all the terms will vanish except for m equals 1. So that gives you a simplified version. You can 00:15:17.580 --> 00:15:26.880 find out what is a l1 that term comes out to be like this and B l1 is basically i A l1. How does 00:15:26.880 --> 00:15:32.160 it help you? You can actually use this similar kind of expression for scattered and the internal 00:15:32.160 --> 00:15:40.140 field. So scattered field is also represented as combination of the vector harmonics, okay. 00:15:40.140 --> 00:15:46.320 Similarly internal fields are also written like this. I am not going into the description but what 00:15:46.320 --> 00:15:51.300 happens after you find out the incident field, the internal field and the scattered field you can 00:15:51.300 --> 00:15:56.880 apply the boundary conditions now. So the boundary conditions say that the tangential component of 00:15:56.880 --> 00:16:02.580 the electric field and the magnetic field are continuous across the sphere boundary. So if you 00:16:02.580 --> 00:16:08.760 consider the radius to be a you can write that E incident plus E scattered minus E internal 00:16:09.360 --> 00:16:19.140 cross r cap equals 0. Similarly, E incident plus e scattered minus E internal cross product with 00:16:19.140 --> 00:16:26.220 r cap so there is a curl, okay is also 0. So this is how you can actually this curl is 00:16:26.220 --> 00:16:32.880 actually giving you what? It is giving you the tangential components, okay. So they become 0. 00:16:32.880 --> 00:16:43.800 Now you can write down Mie coefficients as a size independent. So x equals k naught a, okay. 00:16:43.800 --> 00:16:50.220 So a is the radius. So x is actually containing the information of the radius as well as the 00:16:50.220 --> 00:16:57.360 wavelength of the incident light and you can find out all this coefficient al bl cl dl 00:16:58.080 --> 00:17:07.860 and that helps you to actually compute all this particular fields. So what are the fields? 00:17:07.860 --> 00:17:13.620 External field and external field or scattered field you can find out from this calculations. 00:17:13.620 --> 00:17:19.620 So from this you are also able to find out the amount of scattered light, amount of 00:17:20.940 --> 00:17:26.160 what is not scattered is basically absorbed. So those kind of things you can find out exactly 00:17:26.160 --> 00:17:34.560 for a spherical symmetry, okay. So for spherical particles Mie theory provides exact solution. 00:17:35.520 --> 00:17:43.800 Now this is what we have been looking so far. So if you think of plasmon as a overall picture we 00:17:43.800 --> 00:17:49.440 have seen bulk plasmon, surface plasmon and then particle plasmon. We have seen that for 00:17:49.440 --> 00:17:57.120 the bulk plasmon the condition was that epsilon m omega should be equal to 0. So 00:17:57.120 --> 00:18:03.240 that is where the permittivity of that metal becomes 0. So this is the boundary and that 00:18:03.240 --> 00:18:09.960 happens at plasma frequency omega p, okay. And the value is square root of n e square over 00:18:09.960 --> 00:18:15.660 m epsilon 0. So here you can see all this n is the electron concentration, e is the 00:18:15.660 --> 00:18:20.700 electronic charge, m is the mass of electron and epsilon 0 is the permittivity vacuum, 00:18:20.700 --> 00:18:26.580 okay. So all these parameters are basically fixed. So plasma frequency, bulk plasma 00:18:26.580 --> 00:18:32.880 frequency is also not tunable. Whereas you can go to surface plasmon and this is the condition. 00:18:32.880 --> 00:18:42.060 We have seen that epsilon m omega plus epsilon d equals 0. So the condition is basically epsilon m 00:18:42.060 --> 00:18:47.880 omega equals minus epsilon d. So where they are matching you are able to excite surface plasmon 00:18:47.880 --> 00:18:54.420 resonance. And here the surface plasma frequency actually becomes omega sp over, 00:18:55.140 --> 00:19:01.560 so that is omega sp is equal omega p over square root of 1 plus epsilon d. So you can reduce the 00:19:01.560 --> 00:19:08.040 frequency when you go to surface plasmon. So our aim would be here to find out what is 00:19:08.040 --> 00:19:13.740 the resonance condition for particle plasmon or localized surface plasmon 00:19:13.740 --> 00:19:20.100 and then what will be the surface plasma frequency in this case. So let us look into a 00:19:20.100 --> 00:19:27.360 much more simplified approximation to Mie theory that is basically quasi-static approximation. 00:19:27.900 --> 00:19:34.440 Now what is quasi-static approximation? The name itself suggests it is quasi-static. So for very 00:19:34.440 --> 00:19:40.620 small particles when I say very small, the radius of the particle is much much smaller than the 00:19:40.620 --> 00:19:47.820 wavelength, okay. We can say that the phase of the harmonically oscillating electromagnetic field is 00:19:47.820 --> 00:19:52.740 practically constant over the particle volume because the particle is very small, okay. 00:19:52.740 --> 00:20:00.360 So in that case instead of electrodynamics you are able to use the electrostatics, okay. So you 00:20:00.360 --> 00:20:08.040 can actually think of this particular situation that you have a homogeneous isotropic sphere of 00:20:08.040 --> 00:20:16.320 radius a that is located at origin here, okay. So this is the permittivity of the metal, this is the 00:20:16.320 --> 00:20:22.800 permittivity of the surrounding dielectric, there is the incident electric field e0, radius is a, 00:20:23.520 --> 00:20:32.220 p is the particular point and this is angle theta, okay. So in this case you can think of 00:20:32.220 --> 00:20:39.120 Laplace equation. So del square phi equals 0 and e is nothing but minus grad phi. 00:20:39.120 --> 00:20:44.160 So you can solve Laplace equation to find out the potential in and outside the 00:20:44.760 --> 00:20:52.440 particle. So here also due to azimuthal symmetry you can ignore the phi dependency and you can 00:20:52.440 --> 00:21:01.020 simply take the dependency of r and theta. So again the potential can be you know split into 00:21:01.020 --> 00:21:08.820 variables like R of r, theta of theta and then you solve the theta equation it gives 00:21:08.820 --> 00:21:15.540 you again the Legendre polynomial, so theta equals P l cos theta and when you solve the r 00:21:15.540 --> 00:21:25.320 equation you get r equals small r to the power l or r to the power minus l plus 1. So in that 00:21:25.320 --> 00:21:31.380 way you are also able to write down what is the potential inside that is phi in that is inside 00:21:31.380 --> 00:21:37.320 the sphere and also phi out that is outside the sphere. So these are the two expressions 00:21:37.320 --> 00:21:43.500 from this equations that tells you what is the potential inside and outside the sphere. 00:21:43.500 --> 00:21:49.500 So these are the new coefficients that you have introduced. Now there are certain things 00:21:49.500 --> 00:21:55.620 like phi in should be finite at the origin also when you look at phi out, 00:21:56.580 --> 00:22:07.620 so phi out when it is too far it should be same as the incident electric fields potential assuming 00:22:07.620 --> 00:22:15.300 that the particles effect is no longer present at a far distance. So phi out is E naught z which is 00:22:15.300 --> 00:22:24.180 nothing but minus E naught z you can write it as minus E naught r cos theta as r tends to infinity. 00:22:24.900 --> 00:22:29.460 So if you apply these two conditions you will be able to find out the coefficients, 00:22:29.460 --> 00:22:39.120 so you will be able to find out what is B 1. So B 1 equals minus E naught and B l equals 0 for 00:22:39.120 --> 00:22:45.180 all other cases when l is not equal to 1. So in that case you are able to find out one 00:22:45.180 --> 00:22:52.020 coefficient that is B 1. How about A 1 and C 1? So here also you can say that 00:22:52.020 --> 00:22:58.800 the tangential components of the electric fields are continuous that means C l equals A l equals 00:22:58.800 --> 00:23:04.560 0 for all the cases l is not equals 1. So let us take the tangential components, 00:23:04.560 --> 00:23:13.500 so minus 1 over a doh phi n over doh theta at r equals a that is at this point for inside 00:23:13.500 --> 00:23:19.440 potential and outside potential they should be same. So that gives you this kind of a equation. 00:23:20.100 --> 00:23:25.740 What is the other case that the normal component of the electric field is also continuous. 00:23:26.400 --> 00:23:31.980 Here in quasi-static approximation the electric field magnetic field does not come into the 00:23:31.980 --> 00:23:36.720 picture. So we will be considering about the flux, so the normal component of the flux will 00:23:36.720 --> 00:23:44.280 also be continuous. So you can find out what is the flux here minus epsilon naught epsilon m and 00:23:44.280 --> 00:23:51.780 then you have this one which is e basically doh phi in over doh r and you are calculating at this 00:23:51.780 --> 00:23:58.500 boundary so that is r equals a. Similarly you can put it for this one outside region this one. 00:23:58.500 --> 00:24:05.460 So this gives you this particular expression. So you have two variables A 1 and C 1 and you have 00:24:05.460 --> 00:24:11.700 got two equations now. You solve for it and you can get what are the coefficients A 1 and C 1. So 00:24:12.840 --> 00:24:18.240 mathematics is looking bit messy but it is not that complicated if you are interested you can 00:24:18.240 --> 00:24:24.660 always try this on your own else you can simply take this coefficients from this slide that B 00:24:24.660 --> 00:24:33.360 1 is minus E naught, A 1 is minus 3 epsilon d over epsilon m plus 2 epsilon d times e naught 00:24:33.360 --> 00:24:40.320 and C 1 is basically epsilon m minus epsilon d over epsilon m plus 2 epsilon d times e naught. 00:24:40.320 --> 00:24:44.640 So you have got all three coefficients A 1, B 1 and C 1. So now you are in a 00:24:44.640 --> 00:24:51.360 position to write down what is the electric potential inside and outside the particle. 00:24:51.360 --> 00:25:00.000 So phi in is having A 1 coefficient phi out has got B 1 and C 1 coefficient. So you can put those 00:25:00.000 --> 00:25:05.700 here and this is how it looks like. So this is the potential inside the particle and this is 00:25:05.700 --> 00:25:10.800 the potential that is outside the particle. Now there is something interesting in this particular 00:25:10.800 --> 00:25:17.400 expression of potential outside the particle. So if you see there is a distinct contribution 00:25:17.400 --> 00:25:23.460 coming from the electric field which is the incident electric field that is fine and then 00:25:23.460 --> 00:25:29.580 there is an extra component that is coming into the picture from the particles point of view. 00:25:30.540 --> 00:25:37.860 So if you look here so this looks like there is a dipole and there is a potential 00:25:37.860 --> 00:25:44.520 because of this dipole. So you can actually think of a dipole with a polarizability p 00:25:45.360 --> 00:25:52.260 which is given by this expression. So this term you can take as p dot r over 4 pi epsilon naught 00:25:52.260 --> 00:25:59.520 epsilon d r cube. So when you equate these two you will get that p equals this is the expression 00:26:00.120 --> 00:26:06.660 and this can be written as and this polarization is proportional to the electric field. 00:26:06.660 --> 00:26:14.220 So that constant you can take as polarizability alpha. So alpha is 00:26:15.180 --> 00:26:24.360 the term 4 pi a cube then this one epsilon m minus epsilon d over epsilon m plus 2 epsilon d. 00:26:24.360 --> 00:26:30.600 So this can be written as 3 V that is the volume of the particle. So what is the volume V of this 00:26:30.600 --> 00:26:39.000 particle 4 pi by 3 a cube. So 4 pi a cube can be written as 3 V so this is basically 00:26:39.000 --> 00:26:45.840 the polarizability alpha of the particle. Now with this particular expression you can 00:26:46.740 --> 00:26:51.720 find out what is the resonance condition. So you can see that the polarizability 00:26:51.720 --> 00:26:58.260 will experience a resonant enhancement when the condition 00:26:59.580 --> 00:27:08.640 that the denominator of this is very small. That means when modulus of epsilon m plus 2 epsilon 00:27:08.640 --> 00:27:17.820 d is very small. So you can now epsilon m is basically a complex right because it is a metal. 00:27:18.540 --> 00:27:25.200 So metal in visible wavelength typically they have complex permittivity. So in this condition 00:27:25.200 --> 00:27:30.360 and what is epsilon d that is basically the dielectric one which is a real number. 00:27:30.360 --> 00:27:38.520 So you can think of this as real of epsilon m omega plus 2 epsilon d will be equal to 0. So this 00:27:38.520 --> 00:27:48.360 is a condition for the resonance and it is also known as Frohlich condition So you can understand 00:27:48.360 --> 00:27:55.140 that this is the resonance condition. Now if you consider the metal to be a Drude kind of metal. 00:27:55.680 --> 00:28:01.800 So epsilon m equals 1 minus omega p square over omega square plus i gamma omega where gamma is 00:28:01.800 --> 00:28:08.460 the damping constant and if you try to plot the modulus of this polarizability over a normalized 00:28:08.460 --> 00:28:15.240 frequency the frequency is omega over omega p here. In that case if you take gamma equals 0 00:28:17.100 --> 00:28:22.380 that you will give you a infinite because then this term becomes 0 completely. 00:28:23.040 --> 00:28:28.800 So you will get a asymptote it is going up infinitely but if you consider some finite value 00:28:28.800 --> 00:28:38.160 of gamma then you will get this 0 for gamma equals 0.07 omega p you will get this blue curve 0.1 00:28:38.160 --> 00:28:43.680 omega p you will get this curve and 0.2 omega p you will get this curve. So what you can see is 00:28:43.680 --> 00:28:51.720 that as gamma that is a damping getting increased the q factor of the resonance decreases the 00:28:51.720 --> 00:28:58.200 resonance position more or less remains same but the width of the peaks are getting broader. 00:28:58.800 --> 00:29:06.780 So that is what is more damped. So now coming back to this slide to fill this vacant spot so the 00:29:06.780 --> 00:29:15.060 condition now we have derived that epsilon m omega should be equal to minus 2 epsilon d for resonance 00:29:15.060 --> 00:29:23.940 or you can say epsilon m omega plus 2 epsilon d equals 0 and LSPR frequency is nothing but omega 00:29:23.940 --> 00:29:32.460 p over square root of 1 plus 2 epsilon d ok. So this is how the resonance frequencies have shifted 00:29:32.460 --> 00:29:40.560 from bulk to surface to particle plus bonds. Now coming back to the point that oscillating 00:29:40.560 --> 00:29:48.180 dipoles radiate so you can actually find out the electromagnetic field which is associated with 00:29:48.180 --> 00:29:55.920 an oscillating electric dipole ok. So this is the exact calculation this is not within quasi-static 00:29:55.920 --> 00:30:04.140 approximation this is a exact theory so you can see that H is nothing but this is the expression 00:30:04.140 --> 00:30:10.080 for H what is n? n is the unit vector in the direction of the point of interest and p is the 00:30:11.400 --> 00:30:18.240 dipole moment of that particular dipole. So this two gives you the expression for E and 00:30:18.240 --> 00:30:28.440 H ok. So from this you can see that in the near field the fields are you can say predominantly 00:30:28.440 --> 00:30:37.140 electric because in the near field regime you will get mainly electric fields and in the radiation 00:30:37.140 --> 00:30:45.480 zone that is when k r is much much greater than 1 ok the fields are of the spherical waveform. 00:30:45.480 --> 00:30:52.140 So this from this particular ones if you put this two conditions so in the near zone k r 00:30:52.140 --> 00:30:58.140 is much much lesser than 1, in radiation zone k r is much much greater than 1 you can find out that 00:30:58.140 --> 00:31:06.360 the field in near field is mainly electric whereas in the radiation zone or the far field zone they 00:31:06.360 --> 00:31:14.280 are of the spherical waveform ok. Now coming back to the quasi-static we have seen that we are able 00:31:14.280 --> 00:31:21.600 to obtain alpha that is the polarizability of a particle. Now what do we do with that we are 00:31:21.600 --> 00:31:27.240 able to find out the scattering and absorption cross section using these two simple formulas. 00:31:28.320 --> 00:31:36.840 So scattering cross section will be k to the power 4 over 6 pi modulus of alpha square and 00:31:38.340 --> 00:31:45.420 c abs that is the cross section of absorption will be k imaginary of alpha. 00:31:46.020 --> 00:31:48.240 Now if you see the values here 00:31:49.020 --> 00:31:57.240 scattering is proportional to a to the power 6 and absorption is proportional to a cube. 00:31:57.240 --> 00:32:04.380 So you can understand that for small particles absorption dominate over scattering but as the 00:32:04.380 --> 00:32:11.400 particle size increases scattering quickly gains and then it becomes the major contributing factor 00:32:11.400 --> 00:32:18.240 ok. And extinction cross section is nothing but the summation of the scattering and 00:32:18.240 --> 00:32:25.140 absorption. So extinction means whatever is the amount of light getting lost or extinct. 00:32:25.800 --> 00:32:29.220 So these are the two observations from this particular formula 00:32:30.420 --> 00:32:40.200 fine. So you can also write down like this like if you take a sphere of volume V which has got a 00:32:40.200 --> 00:32:48.360 metal permittivity epsilon given as epsilon 1 plus i epsilon 2 within the quasi-static limit you can 00:32:48.360 --> 00:32:56.160 write a simplified formula for scattering cross section as 9 omega over c epsilon m to the power 00:32:56.160 --> 00:33:03.600 3 by 2 times V and then this particular ratio. So epsilon 2 is basically the imaginary part of the 00:33:03.600 --> 00:33:10.080 metal permittivity and epsilon 1 is the real part of the metal permittivity. So you can also find 00:33:10.080 --> 00:33:17.940 out what is the resonance here. So again if you look into the plot so this is basically extinction 00:33:17.940 --> 00:33:26.640 spectra of a 50 nanometer gold nanosphere. So what is the difference here? So the black curve is when 00:33:26.640 --> 00:33:33.780 the surrounding media is air. In this case epsilon m is the surrounding media. 00:33:35.100 --> 00:33:44.700 So it is not metal it is here it is surrounding media is epsilon m. So if you put epsilon m 00:33:44.700 --> 00:33:51.300 equals 1 you will get this particular graph from this equation and then red one is water 00:33:51.300 --> 00:33:59.700 and n is the refractive index that is 1. 33. So epsilon m will be square of 1.33 that 00:33:59.700 --> 00:34:11.820 is 1.69 I believe and then the last one is for silica that is n equals 1.5. So epsilon m in 00:34:11.820 --> 00:34:18.360 this case will be 2.25. So this is what we can see that you are able to do the refractive index 00:34:18.360 --> 00:34:26.580 sensing using this kind of gold nanosphere because the resonance peak position is getting changed. 00:34:26.580 --> 00:34:36.780 So as the dielectric okay so there is a bit of yeah this should be d okay or you can say this 00:34:36.780 --> 00:34:46.020 is medium I will correct this later on. So this is basically epsilon d okay. You can say it is a 00:34:46.020 --> 00:34:54.480 dielectric or you can also say this is the medium okay. So when epsilon d increases the resonance 00:34:54.480 --> 00:35:02.160 frequency is also from here to here the resonance frequency actually increased yeah no sorry the 00:35:02.160 --> 00:35:08.820 wavelength actually increased it means the resonance frequency actually decreased and your 00:35:09.540 --> 00:35:16.020 cross section of extinction actually increases. So once again this is in terms of wavelength. So 00:35:16.020 --> 00:35:23.460 wavelength is increasing means it is getting red shifted that means the energy is getting reduced. 00:35:25.440 --> 00:35:31.860 You can also apply quasi-static approximation for non-spherical particles something like 00:35:31.860 --> 00:35:40.860 nano rod or nano ellipsoid. So in that case you can have 3 particular axis so 00:35:40.860 --> 00:35:47.580 you can consider this particular case is in ellipsoid so it will have 3 axis a 1, 00:35:47.580 --> 00:35:55.560 a 2 and a 3 these are basically the semi-axis. So this is how the equations are correlated. 00:35:57.120 --> 00:36:05.040 So you can find out polarizability in different directions okay for 00:36:05.880 --> 00:36:13.620 the 3 cases. So if you consider the cross section to be same and only the length to be different so 00:36:13.620 --> 00:36:20.220 there will be 2 different cases so which are shown here. So you can also have longitudinal excitation 00:36:20.220 --> 00:36:28.620 or transverse excitation of the electrons on the surface of such nanoparticles depending on their 00:36:29.700 --> 00:36:35.700 incident electric field polarization. So the light is falling from the top it has got a horizontal 00:36:35.700 --> 00:36:42.600 electric field like this so the electrons will also oscillate along the length of the nano rod. 00:36:43.440 --> 00:36:49.860 So this is the nano rod or nano ellipsoid and you will get one particular kind of polarizability. 00:36:49.860 --> 00:36:56.220 But when the electric field is coming from this direction sorry the light is coming 00:36:56.220 --> 00:37:01.800 from this direction with this electric field oscillating in this direction up and down like 00:37:01.800 --> 00:37:08.580 this then the polarizability will be different. So alpha i gives you the polarizability along the 00:37:08.580 --> 00:37:15.420 3 direction which has got a geometrical factor L i in this particular equation and l i can be 00:37:15.420 --> 00:37:22.080 obtained from here. So as you can see this is a generic case for a sphere L 1, L 2 and L 3 are 00:37:22.080 --> 00:37:30.360 same one third because summation of L i should be equal to 1. So with this you are able to find out 00:37:30.360 --> 00:37:40.380 the polarizability of an ellipsoid and if you plot the absorbance versus wavelength you will 00:37:40.380 --> 00:37:50.040 see that for the 2 cases okay so if you change the aspect ratio that is the ratio of the height 00:37:50.820 --> 00:37:58.920 over the diameter of this okay so as you change the aspect ratio there is a large shift of the 00:37:58.920 --> 00:38:05.160 resonance. First observable thing is that in such a particle there are basically 2 resonance 00:38:05.160 --> 00:38:11.040 peak so one resonance this particular resonance peak corresponds to the surface plus 1 resonance 00:38:11.040 --> 00:38:18.840 along the length of the nanorod whereas this one is along the transverse direction so this 00:38:18.840 --> 00:38:25.920 is called transverse dipolar resonance this is called longitudinal dipolar resonance and when 00:38:25.920 --> 00:38:32.040 you increase the aspect ratio of the particle you see that the transverse peak does not shift that 00:38:32.040 --> 00:38:42.060 significantly whereas the peak of the longitudinal plus 1 undergoes much more red shift red shift 00:38:42.060 --> 00:38:48.300 means shift towards longer wavelength what happens to the energy? Energy reduces okay. 00:38:48.840 --> 00:38:55.260 Now quasi-static approximation is it good throughout? No. For very very tiny particles 00:38:55.260 --> 00:39:01.560 or very very large particles it is a problem so let us look into the case of first very large 00:39:01.560 --> 00:39:07.800 particles where the particle size is more than 100 nanometer. So in those cases you should go 00:39:07.800 --> 00:39:14.160 for Me theory because they provide you an exact solution in form of power series expansion and 00:39:15.120 --> 00:39:22.380 quasi-static are basically the only the first order terms from that expression okay however for 00:39:23.580 --> 00:39:32.340 quasi-static to little larger particles you can actually add some extra terms like you know you 00:39:32.340 --> 00:39:37.860 can add some effects coming from redshift due to retardation okay then broadening 00:39:37.860 --> 00:39:43.860 due to radiative decay and some higher order resonance you can add those terms 00:39:44.700 --> 00:39:51.600 to make your quasi-static approximation little bit more accurate. But always remember for spherical 00:39:51.600 --> 00:39:59.220 particles Me theory provides you the exact solution. So this is what we have seen that if 00:39:59.220 --> 00:40:05.880 you start from 50 nanometer where this is the size of the particle your quasi-static approximation 00:40:05.880 --> 00:40:12.420 is pretty good but slowly as you move towards larger particle up to here 100 nanometer okay 00:40:12.420 --> 00:40:16.800 we will see that there is a redshift there is a spectral broadening so these are the 00:40:16.800 --> 00:40:23.820 effects that comes into picture. So there will be radiative decay that gives you spectral broadening 00:40:23.820 --> 00:40:28.860 there is also redshift due to retardation because when the particle becomes large 00:40:30.360 --> 00:40:37.500 the initial approximation that the electric field over the particle volume does not change 00:40:37.500 --> 00:40:44.880 significantly that does not hold good so there will be a phase lag between the electron movement 00:40:44.880 --> 00:40:49.920 from one end on the others end of the particle because the particle size is large so all these 00:40:49.920 --> 00:40:57.360 effects will be considered and they will try to make your quasi-static theory more inaccurate 00:40:57.360 --> 00:41:05.220 when you try to go to larger particles. Here is an figure that shows you that with 00:41:05.220 --> 00:41:10.680 the size of the particle and change in the permittivity how the resonance wavelength is 00:41:10.680 --> 00:41:19.980 going to change and here is the damping pathways for particle plus 1 means there are two types of 00:41:19.980 --> 00:41:27.180 decay one is by the radiative decay the name itself tells you a radiation through which the 00:41:27.180 --> 00:41:33.300 decay takes place so you will get a photon out of it the other damping may be because of the non 00:41:33.300 --> 00:41:40.620 radiative factors that is it is getting absorbed okay and this is this can be of two types like 00:41:40.620 --> 00:41:48.540 it can be inter-band from one band from the d band to sp band if you go and or it can be also 00:41:48.540 --> 00:41:56.580 intra-band so within the same band you are moving from a lower to higher energy level okay so that 00:41:56.580 --> 00:42:03.060 kind of transition also can give you damping in this particular particles so overall you 00:42:03.060 --> 00:42:09.600 will be able to get broadening because of this. Now if you come to the other end like if you go 00:42:09.600 --> 00:42:16.200 to extremely small particles which are less than 10 nanometer in size then also there is a problem 00:42:16.860 --> 00:42:24.960 like usually for particles which are smaller than the electron mean free path in a metal say in gold 00:42:24.960 --> 00:42:32.460 the mean free path is 42 nanometer so one it means an electron can travel up to 40 or 42 nanometer 00:42:32.460 --> 00:42:38.100 without colliding with another electron. Now if the particle itself is smaller than 42 nanometer 00:42:38.100 --> 00:42:43.920 what will happen before the electrons get collided with another electrons they will actually hit the 00:42:43.920 --> 00:42:50.040 boundary of the particle and that will actually give you additional damping okay so these are 00:42:50.040 --> 00:42:56.640 called additional damping coming from reduced mean free path and that can be also empirically 00:42:57.540 --> 00:43:05.220 associated with some broadening which and there is some simple formula that can correlate the damping 00:43:05.220 --> 00:43:13.980 constant gamma with the actual damping constant plus a is a constant which is usually taken as 00:43:13.980 --> 00:43:22.260 one for isotropic cases vf is the Fermi velocity and R is the radius or the reduced distance of the 00:43:22.260 --> 00:43:28.680 collision. If you are thinking of a thin film or a thin shell you can take that shell dimension or 00:43:28.680 --> 00:43:35.040 thin film dimension as this R and that will give you some additional damping and that you can put 00:43:35.040 --> 00:43:42.840 it back into the Drude formula to get what will happen to this particular resonance. So here is an 00:43:44.340 --> 00:43:51.420 calculation of such line width and defacing time versus resonance 00:43:52.200 --> 00:44:00.480 energy and this has been done for gold as well as silver and is as you can see that this line is 00:44:00.480 --> 00:44:10.740 basically Mie theory and for the particles which are smaller like 40 nanometer, 60 nanometer they 00:44:10.740 --> 00:44:15.300 are very much lying on the Mie theory. Similarly you can also see from here 00:44:16.020 --> 00:44:22.320 okay for extremely small particles that is where the radius becomes one nanometer or so 00:44:22.920 --> 00:44:27.960 that is where you have to remember that the classic electromagnetic theory will no longer hold 00:44:27.960 --> 00:44:35.400 good so you have to go for quantum mechanics to solve it. So there the energy gained by individual 00:44:35.400 --> 00:44:44.280 electrons per incident photon excitation can be written as delta E which is H cross nu over n 00:44:44.820 --> 00:44:50.580 and that should be greater than equal to kBT and N is what N is the absolute number of electrons 00:44:50.580 --> 00:44:57.000 in that particular particle. So with that we understood that there are particle plus 00:44:57.000 --> 00:45:03.360 bonds which are tunable and what are the different damping mechanisms for this and there is another 00:45:03.360 --> 00:45:11.220 sort of plus bond that is also possible which is called void plus bond. So void means if you 00:45:11.220 --> 00:45:17.820 take a sheet of metal and make a whole out of it there also you will see that when there is 00:45:19.920 --> 00:45:27.240 light incident on this they will if the whole is sub wavelength you are able to excite electron 00:45:27.240 --> 00:45:31.860 clouds to move on one side so you will get positive on the other side so this behaves 00:45:31.860 --> 00:45:42.420 like a source of plus bond. So in this case you swap epsilon m and epsilon d so this is 00:45:42.420 --> 00:45:48.000 the metal this is the dielectric permittivity if you swap this in the metal sphere dipole moment 00:45:48.000 --> 00:45:54.120 what you got in the previous one you get this. So in this case the Frohlich condition of the 00:45:54.120 --> 00:46:01.680 resonance also will get changed so it becomes epsilon d plus 2 epsilon m equals 0 or you can 00:46:01.680 --> 00:46:08.280 write epsilon m omega equals minus epsilon d over 2 that means you can also find out what is the 00:46:08.280 --> 00:46:16.200 resonance frequency that comes out to be square root of 2 by 3 omega p that is the resonance 00:46:16.200 --> 00:46:23.400 frequency for void plus bonds are also different. Also there are two cases like this the sphere plus 00:46:23.400 --> 00:46:30.420 bond that you have seen where it is omega B that is the bulk or you can say omega p that is the 00:46:30.420 --> 00:46:38.280 plasma frequency of the bulk metal over square root of 3 here it is different okay there is an 00:46:38.280 --> 00:46:45.480 additional factor of square root of 2 coming into the picture. Now there are two possible hybridized 00:46:45.480 --> 00:46:53.100 modes for these two things to come together and create a shell so this is called a nano shell. 00:46:53.100 --> 00:47:00.780 So in nano shell there are possibilities that they are in the anti-bonding kind of orientation so 00:47:00.780 --> 00:47:06.300 where the outside dipole and the inner dipole are in the opposite direction or you can also 00:47:06.300 --> 00:47:11.940 have this case where the outside dipole so you can see the outside layer plus plus charges then 00:47:11.940 --> 00:47:17.340 out this side also there are negative charges they create one dipole towards the inner side you will 00:47:17.340 --> 00:47:24.900 also have another dipole plus and minus. So here both the dipoles are in the same orientation so 00:47:24.900 --> 00:47:32.700 the overall energy is lowered okay and this is called bonding type of interaction and this is 00:47:32.700 --> 00:47:37.080 called anti-bonding type of interaction. So in this case in this particular two 00:47:38.880 --> 00:47:45.000 cases you will have two hybridized modes the resonance frequency of these modes are given as 00:47:45.000 --> 00:47:52.680 omega plus minus okay so this is how it is obtained. So you have 1 plus minus 1 over 2 L plus 00:47:52.680 --> 00:48:00.180 1 square root of so you can take L equal to 1 and find out what is the first order okay anti-bonding 00:48:00.180 --> 00:48:08.160 mode and bonding mode for this particular cases. So they are also tunable because you can change 00:48:08.160 --> 00:48:14.880 the metal you can change the shell thickness and you can also get a lot of tunability out of this 00:48:14.880 --> 00:48:21.780 void plasmons okay or you can say nano shells void plasmons are this one you can nano shells if you 00:48:21.780 --> 00:48:28.440 remember from the initial lectures they were the ones having the largest tunability. So depending 00:48:28.440 --> 00:48:33.600 on the application you are able to design nano shells that can give you you know that particular 00:48:33.600 --> 00:48:40.380 resonance at that particular operating wavelength okay. So with that we will stop here today and 00:48:41.280 --> 00:48:47.100 in the next lecture we will go into little bit of more details of this resonance effects 00:48:47.100 --> 00:48:52.140 and if you have got any queries regarding to this lecture you can always drop an email at 00:48:52.140 --> 00:48:55.860 this particular email address mentioning MOOC on the subject line. Thank you.