Plasmon resonance and heat generation model in nanostructures
aa r X i v : . [ m a t h . A P ] D ec Plasmon resonance and heat generation model innanostructures ∗ Xiaoping Fang † Youjun Deng ‡ Jing Li § Abstract
In this paper, we investigate the photothermal effects of the plasmon resonance.Metal nanoparticles efficiently generate heat in the presence of electromagnetic radi-ation. The process is strongly enhanced when a fixed frequency of the incident waveilluminate on nanoprticles such that plasmon resonance happen. We shall introducethe electromagnetic radiation model and show exactly how and when the plasmon res-onance happen. We then construct the heat generation and transfer model and derivethe heat effect induced by plasmon resonance. Finally, we consider the heat generationunder plasmon resonance in a concentric nanoshell structure.
Mathematics subject classification (MSC2000):
Keywords: photothermal effect, plasmon resonance, nanoparticle, Maxwell equation,concentric nanoshell
There has been a great deal of interest in recent years in the study of physical property -heat generation by Nanoparticles (NPs) under optical illumination. The optical propertiesof NPs, including both semiconductor [15] and metal nanocrystals [7], have been studiedintensively. When the NPs are illuminated by an incident wave with special frequency, theplasmon resonance can be excited with a strong field enhancement inside and around theNPs. Such effect has a wide range of potential applications in such as near-field microscopy[5, 6], signal amplification, molecular recognition [12], nano-lithography [16], ect. Heat isgenerated when imaginary part of the property of the NP does not vanish. The heatingeffect becomes especially strong under the plasmon resonance conditions when the energyof incident photons is close to the plasmon frequency of the NP. The heat can be usedfor melting the surrounding matrix like ice and polymer [8, 14, 21], as well as for cancerdiagnosis and therapy [10, 11, 13, 22]. ∗ Fang is supported by NSF grants No. NSFC70921001 and No. 71210003. Deng and Li are supportedby NSF grants No. NSFC11301040, † Postdoctoral, Management Science and Engineering Postdoctoral Mobile Station, School of Business;School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P. R. China.Email: [email protected] ‡ Corresponding author. School of Mathematics and Statistics, Central South University, Changsha,Hunan 410083, P. R. China. Email: [email protected], dengyijun [email protected] § Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan 410004,P.R.China. Email: [email protected]
1o mathematically explain the plasmon resonance of NPs and the heat generated byplasmon resonance, there are two models to be considered. Firstly, the Maxwell systemis introduced to explain the process of plasmon resonance. We shall give the exact math-ematical theory on how the plasmon resonance happen when the NPs is illuminated byelectromagnetic wave. We show the asymptotic expansion of electromagnetic fields per-turbed by small inclusions derived in [2, 4]. The first order expansion is quite important tosee the innate of the inclusion. It can be used for the size estimation of the small inclusionand more importantly, it gives us a way to see through when the plasmon resonance happensif the inclusions (NPs), such as gold (Au) NPs and Silver (Ag) NPs, have negative real partin the parameters. We show that when the electric permittivity of the NP is such thata contrast parameter becomes the eigenvalue of an integral operator, plasmon resonancehappen. We then analyze the heat generation and transfer process by strictly deduction.Finally, a typical nanostructure of concentric nanoshell is considered and we show exactlyhow is the plasmon resonance excited by incident light.The organization of this paper is as follows. In section 2, we introduce some preliminaryworks concerning the electromagnetic field model. We present the far field expansion of theelectromagnetic field perturbed by the nanoparticles. In the case that the nanoparticles aresphere shaped, we show exactly how the plasmon resonance happen when Drude model isapplied. In section 3, the heat generation and transferring model is considered. The heattransferring is explicit shown for sphere nanoparticle. Finally, in section 4, we consider aspecial concentric nanoshell structure. The strict mathematical proof is given concerningthe plasmon resonance. To our knowledge, it is the first time to show exactly the plasmonresonance in this concentric nanoshell structure.
Let D be the nanoparticle situated in R with C ,η boundary for some η >
0, and let ( ǫ , µ )be the pair of electromagnetic parameters (permittivity and permeability) of the matrix( R \ D ) and ( ǫ , µ ) be that of the nanoparticle. Then the permittivity and permeabilitydistributions are given by ǫ = ǫ χ ( R \ D ) + ǫ χ ( D ) and µ = µ χ ( R \ D ) + µ χ ( D ) , where χ denotes the characteristic function. In the sequel, we set k = ω √ ǫ µ and k = ω √ ǫ µ . Suppose the nanoparticle D is illuminated by a given incident plane wave with theelectric and magnetic fields ( E in , H in ), which is the solution to the Maxwell equations (cid:26) ∇ × E in = iωµ H in in R , ∇ × H in = − iωǫ E in in R where i = √−
1. We can treat the nanoparticle D as a scatterer in front of the incidentwave. The total fields denoted by ( E , H ), is the solution to the following Maxwell equations: ∇ × E = iωµ H in R \ ∂D, ∇ × H = − iωǫ E in R \ ∂D, [ ν × E ] = [ ν × H ] = 0 on ∂D, (2.1)subject to the Silver-M¨uller radiation condition:lim | x |→∞ | x | ( √ µ ( H − H in ) × ˆ x − √ ǫ ( E − E in )) = 0 , x = x / | x | . Here, [ ν × E ] and [ ν × H ] denote the jump of ν × E and ν × H along ∂D ,namely,[ ν × E ] = ( ν × E ) (cid:12)(cid:12) + ∂D − ( ν × E ) (cid:12)(cid:12) − ∂D , [ ν × H ] = ( ν × H ) (cid:12)(cid:12) + ∂D − ( ν × H ) (cid:12)(cid:12) − ∂D where ·| + ∂D means the limit to the outside of ∂D and ·| − ∂D means the limit to the inside of ∂D .In what follows, we recall the analytic solution to (2.1). Firstly, for k >
0, the funda-mental solution Γ k to the Helmholtz operator (∆ + k ) in R isΓ k ( x ) = − e ik | x | π | x | . (2.2)To proceed, we introduce some vector functional space and some important integrals . Let ∇ ∂D · denote the surface divergence. Denote by L T ( ∂D ) := { ϕ ∈ L ( ∂D ) , ν · ϕ = 0 } thetangential vector space. We introduce the function spaces T H (div , ∂D ) : = n ϕ ∈ L T ( ∂D ) : ∇ ∂D · ϕ ∈ L ( ∂D ) o ,T H (curl , ∂D ) : = n ϕ ∈ L T ( ∂D ) : ∇ ∂D · ( ϕ × ν ) ∈ L ( ∂D ) o , equipped with the norms k ϕ k T H (div ,∂D ) = k ϕ k L ( ∂D ) + k∇ ∂D · ϕ k L ( ∂D ) , k ϕ k T H (curl ,∂D ) = k ϕ k L ( ∂D ) + k∇ ∂D · ( ϕ × ν ) k L ( ∂D ) . For a density φ ∈ T H (div , ∂D ), we define the single layer potential associated with thefundamental solutions Γ k given in (2.2) by S kD [ φ ]( x ) := Z ∂D Γ k ( x − y ) φ ( y ) ds ( y ) , x ∈ R . For a scalar density contained in L ( ∂D ), the single layer potential is defined by the sameway. We also define boundary integral operators L kD [ φ ]( x ) := (cid:0) ν × (cid:0) k S kD [ φ ] + ∇S kD [ ∇ ∂D · φ ] (cid:1)(cid:17) ( x ) , M kD [ φ ]( x ) := p.v. Z ∂D ν ( x ) × (cid:16) ∇ x × (cid:0) Γ k ( x − y ) φ ( y ) (cid:1)(cid:17) ds ( y ) , x ∈ ∂D. There admits the following jump formula on the boundary of D ν × ∇ × S kD [ ϕ ] (cid:12)(cid:12)(cid:12) ± = ( ∓ I M kD )[ ϕ ] . (2.3)Then the solution to (2.1) can be represented as the following E ( x ) = ( E i ( x ) + µ ∇ × S k D [ φ ]( x ) + ∇ × ∇ × S k D [ ψ ]( x ) , x ∈ R \ D,µ ∇ × S k D [ φ ]( x ) + ∇ × ∇ × S k D [ ψ ]( x ) , x ∈ D, (2.4)3nd H ( x ) = − iωµ (cid:0) ∇ × E (cid:1) ( x ) , x ∈ R \ ∂D, where the pair ( φ , ψ ) ∈ T H (div , ∂D ) × T H (div , ∂D ) is the unique solution to (cid:20) M L k D − L k D L k D − L k D M (cid:21) (cid:20) φψ (cid:21) = (cid:20) ν × E in iω ν × H in (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ∂D . (2.5)where M := µ + µ I + µ M k D − µ M k D , M := (cid:18) k µ + k µ (cid:19) I + k µ M k D − k µ M k D . Denote by λ ǫ and λ µ the electric permittivity and magnetic permeability contrasts: λ ǫ = ǫ + ǫ ǫ − ǫ ) and λ µ = µ + µ µ − µ ) . (2.6)We point out that if λ ǫ and λ µ are greater than 1 /
2, or the permittivity and permeability areall positive numbers, then the invertibility of the system of equations (2.5) on
T H (div , ∂D ) × T H (div , ∂D ) was proved in [20].
In what follows, we consider the physical process on the plasmon resonance when thenanoparticle D is illuminated by the incident wave. According to the Drude model theelectric permittivity of the nanoparticle D behaviors as a function of the angular frequency ω , or exactly (see e.g. [18]) ǫ = ǫ ( ω ) = ǫ (1 − ω p ω ( ω + iτ ) ) (2.7)where ω p is the plasma frequency of the bulk material and τ is the width of the resonance.Let D = δB + z , where B is a C ,η domain containing the origin. For a scalar density φ ∈ L ( ∂B ), define the well-known Neumann-Poincar´e operator by K ∗ B [ φ ]( x ) := p.v. Z ∂B ∂ Γ ∂ν ( x ) ( x − y ) φ ( y ) ds ( y ) , (2.8)where ∂/∂ν denotes the normal derivative and p.v. denotes the Cauchy principle value.Denote by G ( x , z ) the matrix valued function (Dyadic Green function) G ( x , z ) = ǫ (Γ k ( x − z ) I + 1 k D x Γ k ( x − z ))then there holds the following far field expansion for the electric field Theorem 2.1 (Theorem 3.8 in [2])
Define the polarization tensors M e := Z ∂B ˜ y ( λ ǫ I − K ∗ B ) − [ ν ] ds (˜ y ) and M h := Z ∂B ˜ y ( λ µ I − K ∗ B ) − [ ν ] ds (˜ y ) (2.9)4 hen there holds the following far field expansion E ( x ) − E in ( x ) = − δ ω µ G ( x , z ) M e E in ( z ) − δ iωµ ǫ ∇ × G ( x , z ) M h H in ( z ) + O ( δ ) . (2.10)Since the eigenvalue of Neumann-Poincar´e operator K ∗ B lies in the span ( − / , /
2] (cf. [3]),polarization tensors M e and M h are well-defined if the material parameters are all positive.However, if the parameters of the NP are not positive numbers then polarization tensorsmay not be well-defined since λ ǫ I − K ∗ B and λ µ I − K ∗ B may not be invertible in this case.It is also shown in [2] that when the electric and magnetic properties of the NPs meet thattheir real parts make λ ǫ or λ µ be the eigenvalue of K ∗ B , the plasmon resonance happen. Weshall see when the parameters obey the Drude and the angular frequency of the incidentwave is well chosen, the plasmon resonance is excited. We shall consider that the nanoparticle D is a sphere shaped inclusion. Suppose the radiusof NP is r NP . The electric permittivity of D obeys the Drude model (2.7). We mention thatthe wavelength, denoted by λ , of the incident wave is much larger than the radius of theNP, i.e., λ >> r NP . Suppose the incident electric field is uniformly distributed, E in = E ,where E is a constant vector representing the amplitude of the incoming light. Due tothe symmetric property of D , we suppose the electric potentials, denoted by u , inside andoutside D have the form u = (cid:26) E · x + E · x | x | x ∈ R \ D, E · x x ∈ D. The scattering part E · x / | x | decays fast as the light travels far away. By introducing thespherical harmonic functions Y m ( θ, ϕ ), m = − , ,
1, we actually have E · x = r X m = − a m Y m , E · x | x | = r − X m = − a m Y m , E · x = r X m = − a m Y m where the coefficients a jm , j = 0 , , r NP X m = − a m Y m + r − NP X m = − a m Y m = r NP X m = − a m Y m ,ǫ ( X m = − a m Y m − r − NP X m = − a m Y m ) = ǫ X m = − a m Y m . By solving the above equations we have a m = 3 ǫ ǫ + ǫ a m , and a m = ǫ − ǫ ǫ + ǫ r NP a m . Thus there holds the following relation E = 3 ǫ ǫ + ǫ E , and E = ǫ − ǫ ǫ + ǫ r NP E . E after the incident light ( E ) illumination reads E = Z D | E | d x = (cid:12)(cid:12)(cid:12) ǫ ǫ + ǫ (cid:12)(cid:12)(cid:12) Z D | E | d x . (2.11)In what follows, we show how does the plasmon resonance happen when the electric permit-tivity of NP obeys the Drude model. Mathematically, we give the definition on when doesthe plasmon resonance happen. Definition 2.2
Let ǫ satisfy (2.7). We call the plasmon resonance happen if the energy ofthe NP induced by the illumination of the incident light blows up in the following way lim τ → τ E = ∞ . Based on the Definition 2.2 and (2.11) we can easily get the following result
Theorem 2.3
Let λ and v be the wave length and speed of the incident light, respectively.If λ = 2 πv √ ω p , then the plasmon resonance happen. Proof . The angular frequency ω is given by ω = 2 πvλ = √ ω p . A direct calculation by using (2.7) gives ǫ = (cid:16) − ω p ω + τ (cid:17) ǫ + i ω p τω ( ω + τ ) ǫ = − ǫ + i √ τω p ǫ + O ( τ ) . Then by (2.11) the energy can be estimated by E = (cid:12)(cid:12)(cid:12) ǫ i √ ω p τ ǫ + O ( τ ) (cid:12)(cid:12)(cid:12) Z D | E | d x = √ ω p τ − Z D | E | d x + O ( τ − ) . We thus have lim τ → τ E = ∞ and plasmon resonance happens. ✷ We mention that in the lossless Drude mode case ( τ = 0), the permittivity ǫ turns tosuch that λ ǫ is the eigenvalue of the Neumman-Poincar´e operator K ∗ D . To explain this, wefirst present a lemma which gives the eigenvalue of K ∗ D with respect to eigenfunctions Y m when D is a sphere. Lemma 2.4 (cf. [1]) Let B be a ball with radius r then we have K ∗ B [ Y m ]( x ) = 16 Y m , | x | = r = r , m = − , , . We thus come to the conclusion since in the lossless Drude mode ǫ = − ǫ , and λ ǫ = ǫ + ǫ ǫ − ǫ ) = 16which is exactly the eigenvalue of K ∗ D when the incident light is uniformly distributed.6 Heat generation and transfer
In this section, we consider the heat generated by NPs under plasmon resonance. Heattransfer in a system with NPs is described by the usual heat transfer equation ρ ( x ) c ( x ) ∂T ( x , t ) ∂t = ∇ ( σ ( x ) ∇ T ( x , t )) + Q ( x , t ) (3.1)where T ( x , t ) is the temperature, ρ ( x ), c ( x ) and σ ( x ) are the mass density, specific heat, andthermal conductivity, respectively. Q ( x , t ) is the heat intensity which represents an energysource coming from light dissipation in NPs. It is shown in [9] that Q ( x , t ) relates to theelectric field by Q ( x , t ) = 18 π ω ℑ m ( ǫ ) | E | where ℑ m ( ǫ ) means the imaginary part of the electric permittivity ǫ , given by Drudemodel and E has been calculated in the last section. Thus we get Q ( x , t ) = 18 π ω ℑ m ( ǫ ) (cid:12)(cid:12)(cid:12) ǫ ǫ + ǫ (cid:12)(cid:12)(cid:12) | E | . Heat transfers through the NP requires quite few time. We thus consider the steady stateof this process. Denote by σ and σ NP the thermal conductivities outside and insider theNP, respectively. Then in the steady state, we have (cid:26) σ NP ∆ T + Q = 0 x ∈ D,σ ∆ T = 0 x ∈ R \ D. Since the heat source is generated from the center of the NP, it spreads uniformly in everydirection due to the uniform thermal diffusion properties of the NP and the matrix. It alsodecays to nothing as | x | →
0. We suppose the temperature inside and outside the NP hasthe form T = ( A − Q | x | σ NP x ∈ D,B/ | x | x ∈ R \ D. By using the transmission conditions we thus have A = 2 σ + σ NP σ σ NP Qr NP , B = r NP Q σ . Denote by V NP the volume of the NP then temperature distribution outside the NP is givenby T = r NP Q σ r = V NP Q πσ r , r = | x | > r NP which is in accordance with [9]. It is seen that the heating effect becomes especially strongunder the plasmon resonance conditions since Q is greatly increased under plasmon reso-nance. 7 ε ε ε ε r r r r Figure 1: Schematic of a concentric nanoshells. ǫ and ǫ are composed of dielectric core. ǫ and ǫ are composed of metallic shell. ǫ is the embedding medium. In this section, we investigate the plasmonic properties of a concentric nanoshell. The plas-mon hybridization model has been used to explain the properties of nanoshell, a tunableplasmonic nanoparticle consisting of a dielectric (silica) core and a metallic (Au or Ag) shell.Fig. 1 gives an example of concentric nanoshell with two layers of metal NPs. This concen-tric nanoshell consists of alternating layers of dielectric and metal, essentially a nanoshellenclosed within another nanoshell, inspiring its alternative name of nanomatryushka (cf.[17]). We shall show how does the plasmon resonance happen in this kind of structure. Wecan also see how sensitive does the inner and outer radius of the shell layer influence theplasmon resonance.From the analysis in the last sections, we should only consider the system in the influenceof the electric field. Suppose once again that the incident wave is uniformly distributed withamplitude E . Define A := { r ≤ r } , A j := { r j < r ≤ r j +1 } , j = 1 , , , A := { r > r } . (4.1)By the symmetric properties of the concentric nanoshell, we suppose the total electric po-tential u has the form u = E · x = r P m = − a m Y m , x ∈ A , E j · x + E j · x / | x | = r P m = − a jm Y m + r − P m = − b jm Y m , x ∈ A j , j = 2 , , , E · x + E · x / | x | = r P m = − a m Y m + r − P m = − b m Y m , x ∈ A The transmission conditions on the interface { r = r j } , j = 1 , , , a j +1 m r j + b j +1 m r − j = a jm r j + b jm r − j ,ǫ j +1 (cid:0) a j +1 m − b j +1 m r − j (cid:1) = ǫ j (cid:0) a jm − b j r − j (cid:1) b m = 0 and a m = a m , m = − , ,
1. By setting λ j = 2 ǫ j +1 + ǫ j ǫ j +1 − ǫ j , j = 1 , , , λ ( a m − a m ) − X j =2 ( a j +1 m − a jm ) = − a m − a m − a m ) (cid:16) r r (cid:17) − λ ( a m − a m ) + X j =3 ( a j +1 m − a jm ) = a m X j =1 ( a j +1 m − a jm ) (cid:16) r j r (cid:17) + λ ( a m − a m ) − ( a m − a m ) = − a m − X j =1 ( a j +1 m − a jm ) (cid:16) r j r (cid:17) − λ ( a m − a m ) = a m Define the matrix P and Υ n by P := λ − − − − r /r ) − λ r /r ) r /r ) λ − − r /r ) − r /r ) − r /r ) − λ Υ := r r r
00 0 0 r . (4.3)Then there holds b m = a m ΞΥ P − e (4.4)where b m := ( b m , b m , b m , b m ) T , e := (1 , − , , − T andΞ = . We also have a m = a m ( − Ξ T P − e + e ) (4.5)where a m := ( a m , a m , a m , a m ) T and e := (1 , , , T . What we concern is the energygenerated in the metal shells. By the analysis before, we have in mind that in plasmonresonance mode, the energy blows up if the imaginary part of material property of the metalparticles goes to zero. To simplify the analysis, in what follows we suppose that the materialproperties of the dielectric core are fixed and ǫ = ǫ = ǫ Let ǫ s = ǫ = ǫ . Then we have λ = λ = 1 − λ = 1 − λ = 2 ǫ s + ǫ ǫ s − ǫ . Under these assumptions, we have P = λ I − K , where I is the identity matrix and K hasthe form K = r /r ) − − − r /r ) − r /r ) r /r ) r /r ) r /r ) (4.6)9udging from (4.4) and (4.5), we set the determinant of P , or λ I − K , to be zero and wehave the equation0 = λ − λ + (cid:0) − r r − + 2 r r − − r r − + 2 r r − − r r − + 1 (cid:1) λ + (cid:0) r r − − r r − + 2 r r − + 2 r r − − r r − + 2 r r − (cid:1) λ + 4 r r − r r − (4.7)which is a forth order equation with respect to λ . It is easy to see that the solution areexactly the eigenvalues of the matrix K . We can actually solve the forth order equationexplicitly by using some tools like matlab, etc. It can be seen that all the four roots of theequation are real. However, we shall not discuss the reason here why all solutions should bereal but the reader who is interested may consult any book concerning the theory of algebraequations (see, e.g., [19]). We also see in this equation that the solution only depends onthe ratios of the radii. Table 1 lists the solutions to four different kinds of ratios.Table 1: Solutions to the fourth order equation (4.7). r : r : r : r λ ǫ s /ǫ r : r : r : r λ ǫ s /ǫ -0.5915 -0.1576 -0.5013 -0.19944 : 5 : 9 : 10 1.5915 -6.3439 3 : 4 : 7 : 8 1.5013 -5.0151-0.8550 -0.0508 -0.8237 -0.06241.8550 -19.6878 1.8237 -16.0206-0.6546 -0.1301 -0.3787 -0.26125 : 6 : 11 : 12 1.6546 -7.6849 3 : 4 : 6 : 8 1.3787 -3.8288-0.8771 -0.0427 -0.7702 -0.08301.8771 -23.4024 1.7702 -12.0547The following theorem gives exactly how the plasmon resonance happen in this concentricnanoshell structure. Theorem 4.1
Suppose ǫ = ǫ = ǫ Let ǫ s = ǫ = ǫ . If the angular frequency ω of theincident light is chosen such that lim τ → ǫ s ( ω ) = ǫ ∗ ( ω ) and there exists a vector v ∗ ∈ R such that P ( ǫ ∗ ) v ∗ = 0 , | v ∗ | = 1 and v ∗ · e = 0 then the plasmon resonance happen in concentric nanoshell. Proof . By the assumptions and Drude model we have ǫ s ( ω ) = ǫ ∗ + iτ ω p ω + O ( τ ) . We then have λ ( ǫ s ) = λ ( ǫ ∗ ) + iτ ω p ω (cid:16) − (2 ǫ ∗ + ǫ ) + 1 ǫ ∗ − ǫ (cid:17) + O ( τ ) .
10e also have that λ ( ǫ ∗ ) is one eigenvalue of K , with corresponding eigenvector v ∗ . Fur-thermore, P ( ǫ s ) = P ( ǫ ∗ ) + iτ ω p ω (cid:16) − (2 ǫ ∗ + ǫ ) + 1 ǫ ∗ − ǫ (cid:17) I + O ( τ ) I In the following we let b := ω p ω (cid:16) − (2 ǫ ∗ + ǫ ) + ǫ ∗ − ǫ (cid:17) . By the assumptions we have P ( ǫ s ) v ∗ = ( iτ b + O ( τ )) v ∗ and thus P ( ǫ s ) − v ∗ = ( − iτ − b − + O (1)) v ∗ . Next, we calculate the energy generated in the metal shell. Denote by E the energy generatedin the metal shell then we have E = Z A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m = − ( ra m Y m + r − b m Y m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x + Z A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m = − ( ra m Y m + r − b m Y m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x By using the orthogonality of the spherical harmonic functions Y m , (4.4) and (4.5) we have E = r − r X m = − | a m | + r − r X m = − | a m | − r − − r − X m = − | b m | − r − − r − X m = − | b m | = (cid:16) r − r | f ( ǫ s ) | + r − r | f ( ǫ s ) | + r − − r − | f ( ǫ s ) | + r − − r − | f ( ǫ s ) | (cid:17) X m = − | a m | where the four functions which depend on ǫ s , i.e., f ( ǫ s ), f ( ǫ s ), f ( ǫ s ), f ( ǫ s ) have the form f ( ǫ s ) := − (0 , , , P ( ǫ s ) − e + 1 , f ( ǫ s ) := − (0 , , , P ( ǫ s ) − e + 1 f ( ǫ s ) := (1 , , , P ( ǫ s ) − e , f ( ǫ s ) := (1 , , , P ( ǫ s ) − e . Define F = − − −
10 0 0 − r r r r then ( f ( ǫ s ) , f ( ǫ s ) , f ( ǫ s ) , f ( ǫ s )) T = F P ( ǫ s ) − e + (1 , , , T . Since we have det( F ) = r ( r − r ) = 0, with out loss of generality we suppose (1 , , , v ∗ = 0. Since all the fourterms in the expression of E are positive terms, we only analyze the term which contains f ( ǫ s ) (otherwise, if (0 , , , v ∗ = 0 then we consider f ( ǫ s ) and so on). Let v ∗ , v ∗ , v ∗ and v ∗ be the eigenvectors of K and λ ( ǫ ∗ ), λ ∗ , λ ∗ , λ ∗ be the corresponding eigenvalues. Then e = v ∗ e · v ∗ + v ∗ e · v ∗ + v ∗ e · v ∗ + v ∗ e · v ∗ . Direct calculations give P ( ǫ s ) − e = ( λ ( ǫ s ) − λ ( ǫ ∗ )) − v ∗ e · v ∗ + X j =2 ( λ ( ǫ s ) − λ ∗ j ) − v ∗ j e · v ∗ j = − iτ − b − v ∗ e · v ∗ + O (1) . τ | f ( ǫ s ) | = τ |− iτ − b − (1 , , , v ∗ e · v ∗ + O (1) | = τ − b − | (1 , , , v ∗ | | e · v ∗ | + O (1) . By the assumptions (1 , , , v ∗ = 0 and e · v ∗ = 0 we finally havelim τ → τ E = ∞ which completes the proof by using Definition 2.2. ✷ We make a short remark here. We have seen in Table 1 that there are four possiblefrequencies, according to Drude model, of incident waves which can be used to excite theplasmon resonance. The concentric nanoshell structure, in the mean time, only containstwo metal shells. Thus if the number of metal shells and dielectric cores is increasing, moredifferent frequencies of incident lights can be absorbed and turned to a large amount of heat.The photothermal effects can be used for a variety of imaging and therapeutic applications.
We studied the heat generation and transferring model in the presence of plasmon resonancewhen the NPs are illuminated by incident waves. For a electromagnetic wave illumination,the first order expansion in terms of polarization tensors was presented, and the key pointfor inducing plasmon resonance emerged. We showed strictly how the plasmon resonancehappen when the nanoparticle is sphere shaped and obeys the Drude model. We investigatedthe heat generation and transferring for spherical nanoparticle. The photothermal effect isgreatly enhanced under plasmon resonance. For a concentric nanoshell structure, we provedthat the plasmon resonance happen when the frequency of the incident waves is well chosen.Future works will be focused on the interaction of the nanoparticles under plasmon resonancewhich is not only a very important physical problem but also a great mathematical problem.
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