Size-Independent Parameter for Temperature-Dependent Surface

Aug 24, 2016 - A red shift and band broadening of SPR is observed with increase in temperature for all sizes with significant shift for dimer of radii...
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Size-Independent Parameter for Temperature-Dependent Surface Plasmon Resonance in Metal Nanoparticles Muni Raj Maurya and Vijaykumar Toutam* Quantum Phenomena and Applications Division, CSIR-National Physical Laboratory, Dr. K. S. Krishnan Marg, New Delhi 110012, India ABSTRACT: Temperature (T)-dependent optical response of gold nanoparticle (NP) dimer of radii, 5−60 nm surrounded by air medium is simulated and studied using finite element method (FEM). Temperature-dependent damping parameter from Drude-Lorentz dielectric model having contribution from different scattering mechanisms and volume expansion is taken into consideration for studying surface plasmon resonance (SPR) behavior. A red shift and band broadening of SPR is observed with increase in temperature for all sizes with significant shift for dimer of radii 40−60 nm. A size-independent parameter is extracted from the study, showing similar temperature-dependence, irrespective of dimer size, and can be used as an important parameter in temperature sensing.

1. INTRODUCTION Plasmonics, the study of electromagnetic field interaction with free electrons of metal, has seen an exponential growth in last two decades with huge potential application in nanomedicine,1 biosensing,2,3 nanophotonics,4 photovoltaics,5 optical storage,6 and so on. Optical properties of metal thin films (MTF)/ nanoparticles (MNP) is of great interest as it shows strong scattering and increased absorption for different spectrum of electromagnetic radiation.7 This optical response termed as surface plasmon (SP) in MTF or localized surface plasmons (LSP) for MNPs, is attributed to frequency dependent dielectric function of metals and undergoes resonance at a particular frequency, called as surface plasmon resonance (SPR). The dielectric constant of material surrounding this free electron system plays a significant role in enhancing and positioning of SPR peak. The corresponding waves which propagate under the resonance at metal dielectric interface are sub wavelength in nature and are called surface plasmon polaritons. SPs also show dependence on temperature due to the corresponding variation of dielectric function of the metal and immediate surrounding medium. Free electrons in a metal on interaction with electromagnetic wave undergo forced and damped harmonic oscillations. Drude has given a metal dielectric function based on only intraband transition εD(ω),8,9 later, modified by Lorentz considering the contribution from bound electrons, denoted as interband transition εib(ω). Hence, Drude-Lorentz dielectric function of metal, due to both interband and intraband transition is given by10 ε(ω) = εib(ω) + εD(ω)

compared to vibrational energy, it has minimal or no effect on plasmon absorption with temperature.11 Thus, dielectric function due to interband transition εib(ω) is considered to be temperature-independent and is given by12 εib(ω) = εbulk (ω) − εD(ω , T0)

where value of εbulk(ω) for gold is taken from the work done by Johnson and Christy,13 and εD(ω,T0) is given by eq 3 at room temperature, T0. Temperature-dependent dielectric function due to intraband transition εD(ω) is given by14 εD(ω , T ) = 1 −

ωp2(T ) ω 2 + iγ(T )ω

(3)

ωp(T)12 is plasma frequency, γ(T)15 is damping parameter and their temperature-dependence is given by ωp(T ) =

4πne 2 m × (1 + β(T )(T − T0))

γ(T ) = γe_e(ω , T ) + γe_p(T ) + γe_s(T )

(4) (5)

where γe_e(ω,T), γe_p(T), and γe_s(T) are temperaturedependent electron−electron (e_e), electron−phonon (e_p), and electron−surface (e_s) scattering parameters, respectively. β(T), coefficient of thermal expansion is also temperaturedependent given by

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Received: June 10, 2016 Revised: July 21, 2016 Published: August 24, 2016

Gold has two interband transitions at λ ∼ 470 and 330 nm, respectively. As interband transition energy is very high © 2016 American Chemical Society

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DOI: 10.1021/acs.jpcc.6b05847 J. Phys. Chem. C 2016, 120, 19316−19321

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The Journal of Physical Chemistry C β (T ) =

192ρKB r0 ⌀(16ρ − 7TKB)2

(6)

where ρ, ⌀, and r0 are the parameters of Morse potential to describe potential of interatomic interaction.16 Different temperature-dependent parameters mentioned above are due to Fermi distribution broadening,17 lifetime broadening,18,19 lattice expansion,20 and increased surface scattering with surface area.21 These mechanisms determine density of states and electron distribution in metals, affecting the intraband transition. Watson in 1970 figured out that the energy shift with lattice expansion is due to lowering and narrowing of sp band and d band.20 Several groups have also studied temperature effect on SPR in metal nanoparticles.22−25 Temperature-dependence of SPR is widely studied for its desirable application in thermal sensing. In context with this, several shapes and different-sized NPs are studied. Various SPR-based temperature sensors were reported, using crystal fiber filled with silver nanowires,26 gold NP incorporated oxides,27 polymer-functionalized gold nanoprisms,28 dielectricloaded plasmonic waveguide-ring resonators,29 and so on. SPR dependence on temperature for gold nanoparticle in silica matrix form 17−915 °C was studied by Yeshchenko et al.12 Red shift, along with broadening of SPR band is observed in gold NP with increase in temperature. One of the challenges in terms of temperature sensitivity lies in size of NPs and its monodispersity. In the present work, we carried out simulation on a metal nanoparticle dimer for temperature-dependent surface plasmon resonance using finite element method. Several scattering mechanisms along with lattice expansion are considered for temperature-dependent SPR shift and showed that sizeindependent parameter can be extracted to monitor the temperature. The behavior of this parameter with temperature is same for all sizes, ranging from 5 to 60 nm, and has the potential to be used in thermal sensing.

Figure 1. Schematic for simulation geometry. Dimensions and boundary conditions used in optical analysis of gold dimer.

EMFD. Wavelength sweep from 400 to 700 nm with interval of 10 nm is done for different temperatures. Free deformation is used for air and prescribed deformation due to thermal stress, u in x direction and v in y direction under moving mesh is used for the dimer. Equation for heat transfer used in simulation is given as ρCpμ∇T = ∇(K ∇T )

(7)

where Cp is heat capacity at constant pressure, K is thermal conductivity, ∇T is local temperature gradient, μ is conductance, and ρ is density. These parameters for the elements used in simulation, for example, air and gold are taken from built-in library, provided as piecewise and analytic function in comsol. The wave equation in EMFD used for gold nanoparticles is given below. 1 ∇ × (∇ × E) − ko 2εrE = 0 μr (8) where μr is the relative permeability and εr is the relative permittivity. The total absorbed energy is calculated by integrating the energy loss in dimer.31

2. THEORETICAL METHOD Optical response of gold (Au) dimer is analyzed by simulations using comsol multiphysics software. Physics used for simulation is electromagnetic wave in frequency domain (EMFD). To study further the effect of temperature, structural mechanics due to thermal stress (SMTS) and moving mesh physics in stationary domain is clubbed with EMFD physics. Perfectly matched layer (PML) is used to avoid reflection and to confine the geometry. Scattering boundary condition is applied to make boundary transparent for a scattered wave. The boundary condition is also transparent for an incoming plane wave. Schematic for geometry and conditions used in simulation are shown in Figure 1. Gold nanoparticle dimer with interparticle separation (s) of 2 nm is surrounded by air and whole geometry is enclosed in a square shape PML of thickness 250 nm. Scattering boundary condition is applied at the outer boundaries of PML. Far field domain consists of air and dimer. Mesh is free triangular for entire geometry with mesh element of maximum size, 2 nm for dimer, and extremely fine mesh for the rest of domains as shown in Figure 1. Constant temperature is applied at all the boundaries enclosing air and gold dimer for uniform temperature and to avoid thermal gradient. Temperature-dependent structural deformation is applied on dimer with fixed point constraint at the bottom. Moving mesh30 is used to move the mesh as a function of displacement computed by thermal stress of dimer in SMTS physics coupled with

3. RESULTS AND DISCUSSION Figure 2 shows simulation for electric field enhancement due to localized surface plasmon for a single NP and a dimer with two different separations, along with a plot showing frequency dependent evanescent field for varied separation. The incident EM radiation Eoe−iky, with Eo ∼ 1 V is plane polarized, propagating in the y direction and polarization in the x direction for all cases. Electric field enhancement by single NP of radius (r) 20 nm is shown in Figure 2a. Electrons oscillate along with electric field and emit a scattered field in the direction of oscillation. The intensity of scattered field is higher than incident EM radiation and extends up to few nm, depicted by a yellow region around NP with corresponding scale bar in Figure 2a. No such oscillations occur in the direction perpendicular to electric field represented by small black regions, above and below the NP called as “null points”. Compared to single NP excitation, large field enhancement is observed in the region of separation for dimer as shown in Figure 2b,c. The EM field induces two in-phase dipolemoments which reinforce each other resulting in strong interparticle coupling of charges and large field enhancement in the gap called as “hot spot”. Intensity scale in Figure 2b,c for a dimer of radius 20 nm, with separation 4 and 2 nm, respectively, shows that evanescent field in the hot spot nearly 19317

DOI: 10.1021/acs.jpcc.6b05847 J. Phys. Chem. C 2016, 120, 19316−19321

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The Journal of Physical Chemistry C

Figure 2. Field enhancement due to LSP and interparticle coupling of charges. (a) Enhancement in the electric field due to surface plasmon by single Au nanoparticle of radius 20 nm. (b, c) Enhancement in the electric field for dimer of Au NP with radius 20 nm, separated by 4 and 2 nm, respectively. (d) Frequency-dependent evanescent field plot for different interparticle separation of dimer with inset showing the exponential decay of electric field at the center of hot spot under resonance.

Figure 3. Temperature-dependent SPR for 60 nm NP dimer with 2 nm separation. (i) Variation in the SPR peak (WSPR) for the dimer with increase in temperature from 200−600 K (left y-axis). (ii) Variation of elelctric field intensity (V/m) at resonance with temperature (right y-axis). Surface plasmon simulation for electric field intensity of dimer at the hot spot for different temperatures is shown to the right in figure. A red shift in WSPR and increase in electric field with temperature is observed.

enhancement at resonance with increase in separation is shown in inset of Figure 2d. Figure 3 shows shift in surface plasmon resonance frequency (left) and variation of electric field intensity at the center of hot spot (right) with temperature, for 60 nm dimer and a separation of 2 nm. A red shift of resonance peak with increase in temperature is observed for polarization parallel to dimer’s separation axis as shown in Figure 3. Considering the real part of dielectric function and SPR condition,22 ε′(ω,T) = −2εm, the SP resonance frequency ωSPR(T) for NPs much smaller than wavelength of light is given by

doubles with decrease in interparticle distance. The dimer system maintains the intraparticle charge neutrality. As the separation decreases charge accumulates in region near the gap and has a spatial width of order r × s 32 along the surface of sphere. Spatial width decreases with decrease in interparticle separation leading to strong interparticle coupling of charges, resulting in enhancement of evanescent field in the hot spot. Variation of evanescent field at the center of hot spot with varying frequency is plotted in Figure 2d for separation ranging from 1 to 6 nm. When the oscillating frequency of electron gas matches with incident radiation frequency, the system undergoes resonance. Electric field enhancement reaches its maximum at resonance for given NP size and decays exponentially with increase in interparticle separation given by, y = Aoe(−s/τ) + C,33 where y is hot spot intensity and Ao is incident EM wave intensity. S and τ are separation distance and decay constant, respectively. Exponential decay of field

ωSPR (T ) =

ωp2(T ) 1 + 2εm(T ) + εib′

− γ 2(T ) (9)

Considering eq 9, shift in the resonance frequency with temperature depends on plasma frequency, damping parameter 19318

DOI: 10.1021/acs.jpcc.6b05847 J. Phys. Chem. C 2016, 120, 19316−19321

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The Journal of Physical Chemistry C and dielectric function of medium. With air as surrounding medium, change in permittivity with temperature is negligible and is taken as constant. Since the total number of free electrons in NP is temperature independent,34 electron density decreases with increase in temperature due to volume expansion. This decreases the plasma frequency given by eq 4. The damping parameter, γ is also temperature-dependent. Electron−electron scattering γe_e(ω,T) shows week dependence on temperature due to frequency dependent energy term ℏω (see eq 10), whose value is large compared to the temperature-dependent term KBT in the equation.35,36 γe_e(ω , T ) =

π 4 £Δ ⎡ ⎢(KBT )2 + 6ℏEf ⎣

ℏω 2π

Table 1. Constants Used for Simulating Effect on Surface Plasmon Resonance of Gold Dimer Due to Different Scattering Mechanisms parameter

2⎤

{ }

⎥ ⎦

(10)

where kB is the Boltzmann constant, ℏ is the planck’s constant divided by 2π, Ef is the Fermi energy, £ is the Fermi-surface average for the scattering probability, Δ is the fractional Umklapp scattering coefficient. Following the Debye model, temperature-dependent damping due to electron−phonon scattering γe_p(T)37 is given by ⎡2 4T 5 γe_p(T ) = γ0⎢ + 5 θD ⎣5

∫0

θD/ T

⎤ z4 dz ⎥ e −1 ⎦ z

Fermi-surface average for the scattering probability fractional Umklapp scattering coefficient surface scattering parameter

symbol

value

£

0.55

Δ

0.77

A

0.25

Fermi velocity

υF

1.4 × 106 m/s

damping constant at To

γ0

0.07 eV

Debye temperature

θD

170 K

Fermi energy

Ef

5.53 eV

refs Lawrence et al.36 Lawrence et al.36 Berciaud et al.40 Berciaud et al.40 Ashcroft and Mermin41 Ashcroft and Mermin41 Ashcroft and Mermin41

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where γ0 is the damping constant at room temperature and θD is Debye temperature for gold. Temperature-dependent damping parameter due to electron−surface scattering γe_s(T)38 is given by υ γe_s(T ) = A F R (T ) (12) Damping due to surface scattering becomes more prominent for NP size, r < 10 nm.22 Several values are attributed to surface scattering parameter “A”, based on modeling physics and shape of the nanoparticle.39 Considering the spherical shape and diffusive scattering, A is taken as 0.25.40 υF is the Fermi velocity for bulk gold. Radius after volume expansion, R(T) is given by R(T ) = R 0(1 + β(T )(T − T0))1/3

Figure 4. Size-independent parameter for temperature-dependent surface plasmon resonance. Variation of the ratio WSPR/γ with temperature for dimer of radius ranging from 5−60 nm and separation 2 nm along with curve fitting. This parameter is very much linearly dependent on temperature above To. Inset shows dependence of SPR frequency on temperature.

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separation of 2 nm. At a given temperature and for different dimer size, the magnitude of FDR lies approximately at same value compared to the value of WSPR as shown in the inset. From inset, SPR frequency undergoes red shift with increase in size as the damping due to electron−surface scattering decreases proportionally (see eq 12). Whereas, FDR in the main plot has no dependence on size of the dimer due to compensation of the change in WSPR by damping parameter γe_s(T). Curve fitting of the FDR data shows that the variation of FDR with temperature for a given size has 1/T dependence. Due to negligible contribution from damping due to electron− electron scattering γe_e(ω,T) as it is mostly frequency dependent, the major contribution of damping is from electron−phonon scattering and electron−surface scattering. Considering eq 11 for e_p scattering and for θD/T < 1, the ez term can be written as ez + 1 and the simplified damping term due to e_p scattering has linear dependence on temperature given by

14

Compared to the magnitude of damping parameter (∼10 ), plasma frequency is large (∼1016). Thus, a decrease of SPR frequency with temperature is due to decrease in plasma frequency as given by eq 9 and damping parameter contributes much in the broadening of spectrum. Red line (left axis) in Figure 3 shows decrease of SPR frequency with temperature for dimer of radius 60 nm and is linearly dependent. With increase in temperature, separation between NPs decreases due to lattice expansion and the dimer nearly touch at 600 K. Surface plasmon simulation for electric field intensity of dimer at the hot spot for different temperatures is shown to the right in Figure 3. As a result electric field intensity in the gap increases due to strong interparticle coupling of charges. Exponential increase of electric field intensity at the center of hot spot with increase in temperature is observed at resonance as shown by green line in Figure 3 (right axis). Constants used for simulation of various damping parameters, γe_e(ω, T), γe_p(T), and γe_s(T) are listed in Table 1. Figure 4 shows the behavior of SPR frequency to damping ratio (FDR) with temperature and inset showing behavior of WSPR vs T, for dimer of radius ranging from 5−60 nm and a

⎡2 T⎤ γe_p(T ) = γ0⎢ + ⎥ θD ⎦ ⎣5 19319

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The Journal of Physical Chemistry C γe_p(T) being in the denominator, variation of FDR with temperature should have 1/T dependence. This is in agreement with the behavior of FDR with temperature as shown by curve fitting in Figure 4 (cyan line). Hence, the major contribution for the behavior is due to damping parameter γe_p(T). For values above room temperature, the change in FDR with temperature is mostly linear with a slope ∼0.1/K (magenta line). This linear dependence of FDR on temperature provides easy calibration and enhanced sensitivity. Independence of FDR for various nanoparticle size makes the measurement easy. Hence, FDR can be used as a desirable parameter over SPR frequency for temperature sensing.

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4. CONCLUSIONS Temperature-dependent optical response of gold dimer is studied using finite element method. Behavior of electric field in the hot spot and surface plasmon resonance is analyzed with change in temperature. A red shift is observed with increase in temperature and is in good agreement with literature. Volume expansion is considered to dominate the overall optical response of metal NP dimer. A size-independent parameter (FDR) is proposed for thermal sensing. The behavior of the parameter with temperature and its size independence is explained in terms of different scattering mechanisms. Sensitivity and resolution of the measurement can be increased further by functionalization of NPs, coating with temperaturesensitive molecules, and core−shell structure.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +91-11-45608359. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Dr. Rina Sharma and Dr. Prathap Pathi for their support. Dr. Toutam acknowledges DST for the simulation work done under the support of GAP Project No. 123532. Mr. Muni Raj acknowledges Council of Scientific and Industrial Research (CSIR) and Academy of Scientific and Innovative Research (AcSIR) for fellowship and academic support.

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REFERENCES

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