A method for calculating the absorption and scattering characteristics of nanoparticles at the quantum scale

Abstract

The present invention discloses a method for calculating the absorption and scattering characteristics of nanoparticles at the quantum scale. A nanoparticle model is established in finite element software: from the outside to the inside, there are a perfectly matched layer, an external environment dielectric layer, a weak form computational domain, and a metal particle, where the position of the computational domain added to the weak form computational domain is selected to be in the middle of the external environment dielectric layer and the metal particle region; then the material parameters are set; secondly, the integration region for calculating the absorption factor is set: the weak form computational domain and the metal particle region are set as the integration region for calculating the absorption factor; finally, calculations are performed and the results are exported: a parametric sweep is added under the research module, the change ranges of the incident light wavelength and the radius of the metal nanoparticle are set and the calculations are started, a one-dimensional plot group is selected in the results to plot the absorption factor graph; a two-dimensional plot group is selected to plot the electric field distribution graph.

Description

Technical Field

[0001] The present invention belongs to the field of nano-material simulation, and particularly relates to a method for calculating the absorption and scattering characteristics of nano-particles at the quantum scale. Background Art

[0002] The local surface plasmon resonance (LSPR) effect of noble metal nano-particles can enhance their absorption effect on visible light and obtain a huge local electric field enhancement. This makes it have important application values in aspects such as solar photovoltaic cells and surface-enhanced Raman scattering. However, with the development of nano-particle preparation technology, the sizes of various optical devices are gradually decreasing. When the size of the nano-structure decreases to be close to the mean free path of the electrons in the material, there is a difference between the predicted value of the classical theory and the experimental observation value, which indicates that the classical electromagnetic theory will no longer be applicable. Therefore, it is necessary to establish a modified model at the quantum scale for calculation.

[0003] Under the irradiation of incident light, the free electrons of noble metal nano-particles will oscillate, and then form the local surface plasmon resonance effect, thereby causing strong optical absorption and scattering effects in the far field, and generating a strong electric field enhancement in the near field, that is, around the nano-particles. For example, Wang Mingli et al. (Wang Mingli, Zhu Yanying, Wei Yong, Zhang Le. Theoretical simulation of the local electric field enhancement by rice-shaped silver nano-particles [J]. Infrared and Laser Engineering, 2017, 46(02): 184-190.) used the FDTD algorithm to systematically study the influence of the shape and distance of different combinations of rice-shaped silver nano-particles on the local electric field. It was found that when the polarization direction of the incident light is parallel to the long axis, for a single rice-shaped nano-particle with a long axis of 300 nm and a short axis of 36 nm, its local electric field enhancement effect is the best. By changing the gap between the particles, it was found that as the distance increases, the local electric field strength first increases and then decreases, and when the spacing is 2 nm, their enhanced electric fields all reach the maximum value.

[0004] However, in recent years, with the rapid development of nano-particle preparation technology, the structures of various optical devices are more precise. When the size of the nano-structure decreases to be close to the mean free path of the electrons in the material, the dielectric function obtained from the macroscopic material cannot be applied to the calculation, and it is necessary to consider the activities of the electron states as the plasmon carriers, which has to consider the non-locality of the electrons themselves. And when turning to electrons, many problems can no longer be accurately calculated by classical electrodynamics means. For the extreme near field, it is necessary to perform quantum correction on the dielectric constant. When the nano-gap is shortened to the point where the electron wave functions can overlap, the influence brought by the charge transfer also needs to be considered. Summary of the Invention

[0005] To solve the technical problem of large errors between the results of the classical electromagnetic theory model and experiments when the particle size is reduced, the present invention provides a method for calculating the absorption and scattering characteristics of nanoparticles at the quantum scale.

[0006] The technical solution for achieving the object of the present invention is as follows:

[0007] A method for calculating the absorption and scattering characteristics of nanoparticles at the quantum scale, comprising the following steps:

[0008] Step 1, establish a nanoparticle model in the finite element software COMSOL:

[0009] From the outside to the inside are the perfectly matched layer, the external environment dielectric layer, the weak form computational domain, and the metal particle, where the weak form computational domain is an added computational domain and its position is selected in the middle of the external environment dielectric layer and the metal particle region;

[0010] In the weak form computational domain, the control equation is:

[0011] The weak expression is:

[0012] where test() is the test function in COMSOL, ph x 、ph y 、ph z respectively represent the electric field intensities in the X, Y, and Z directions in the weak form computational domain; V is the volume; k is the propagation constant, emw.epsilonrxx, emw.epsilonryy, emw.epsilonrzz respectively represent the xx, yy, zz components of the relative permittivity; E x 、E y 、E z respectively represent the electric field intensities in the X, Y, and Z directions in the whole space; r 0 is the correction constant;

[0013] At the same time, a weak expression is added in the metal particle region, and the weak expression is:

[0014] -test(ph)*(-ph-dperp / r 0 *(rhob-rhom)) / dperp

[0015] where ph is the electric field intensity in the weak form computational domain; rhob is the surface electric field intensity of air, rhom is the surface electric field intensity of the metal; dperp is the Feibelman d parameter;

[0016] Step 2, set the material parameters, including the relative permeability, conductivity, and relative permittivity;

[0017] Step 3, set the integration region for calculating the absorption factor:

[0018] Set the weak form calculation domain and the metal particle region as the integration region for calculating the absorption factor;

[0019] Step 4, perform calculations and export the results:

[0020] Add parametric sweep under the study module in COMSOL, set the variation range of the incident light wavelength and the variation range of the radius of the metal nanoparticles and start the calculation. In the results of COMSOL, select the one-dimensional plot group, and select the wavelength as the X-axis variable of the image, and the absorption factor Q as the Y-axis variable abs , draw the absorption factor diagram; select the two-dimensional plot group, and select the X-Y plane as the cross-section of the image and the electric field mode as the variable to draw the electric field distribution diagram.

[0021] Compared with the prior art, the remarkable advantages of the present invention are:

[0022] (1) By introducing quantum corrections to handle the plasmon calculation at the quantum scale, and by establishing a local effective model to replace the non-local problem, it is possible to achieve results closer to quantum calculations. Description of the Drawings

[0023] Figure 1 It is a schematic diagram of the classical model

[0024] Figure 2 It is a schematic diagram of the modified model

[0025] Figure 3 It is the Feibelman d parameter diagram of silver

[0026] Figure 4 It is the absorption factor diagram of spherical Ag nanoparticles with different radii under the classical model

[0027] Figure 5 It is the absorption factor diagram of spherical Ag nanoparticles with different radii under the modified model

[0028] Figure 6 It is the electric field distribution of the modified model at 220 nm (a) and the classical model at 280 nm (b) Detailed Embodiments

[0029] The following further introduces the present invention in combination with the drawings and specific embodiments.

[0030] A method for calculating the absorption and scattering characteristics of nanoparticles at the quantum scale according to the present invention includes the following steps:

[0031] Step 1, establish a nanoparticle model in the finite element software COMSOL.

[0032] The schematic diagram of the existing common classical model is as follows Figure 1 . From the outside to the inside, there are a perfect matching layer, an external environmental dielectric layer, and metal particles. The schematic diagram of the modified model of the present invention is as follows Figure 2 . From the outside to the inside, there are a perfect matching layer, an external environmental dielectric layer, a weak form computational domain, and metal particles. The weak form computational domain is the computational domain added in the present invention, and its position is selected in the middle of the external environmental dielectric layer and the metal particle region, and the thickness is set to 20 nm.

[0033] In the weak form computational domain, the control equation is:

[0034] The weak expression is:

[0035] Among them, test() is the trial function in COMSOL, and ph x , ph y , ph z respectively represent the electric field strengths in the X, Y, and Z directions in the weak form computational domain; V is the volume; k is the propagation constant, k = 2π / λ, where λ is the wavelength; emw.epsilonrxx, emw.epsilonryy, emw.epsilonrzz respectively represent the xx, yy, and zz components of the relative dielectric constant; E x , E y , E z respectively represent the electric field strengths in the X, Y, and Z directions in the entire space; r 0 is the correction constant, with a value of 2 nm. The coordinates are in spherical coordinates, and the origin is at the center of the nanoparticle.

[0036] At the same time, a weak expression is added in the metal particle region, and the weak expression is:

[0037] -test(ph)*(-ph-dperp / r 0 *(rhob-rhom)) / dperp

[0038] Among them, ph is the electric field strength in the weak form computational domain; rhob is the surface electric field strength of the air, rhob = up(E x )*n x +up(E y )*n y +up(E z )*n z , up() is the upwind approximation expression in COMSOL, and n x , n y , n zRespectively represent the normal vectors in the X, Y, and Z directions; rhom is the surface electric field strength of the metal, rhom = down(E x )*n x +down(E y )*n y +down(E z )*n z , down() is the lower neighbor estimation expression in COMSOL; dperp is the Feibelman d parameter, which has different parameter values according to different metal materials.

[0039] Step 2: Set the material parameters.

[0040] Set the relative magnetic permeability, conductivity, and relative permittivity parameters of the material.

[0041] Step 3: Set the integration region for calculating the absorption factor

[0042] Set the weak form calculation domain and the metal particle region as the integration region for calculating the absorption factor. The formula for the absorption factor is: where Q abs is the absorption factor, C abs is the absorption cross section, and r is the equivalent volume radius of the metal particle.

[0043] Step 4: Perform the calculation and export the results

[0044] Add a parametric sweep under the study module in COMSOL, set the change range of the incident light wavelength and the change range of the metal nanoparticle radius, and start the calculation.

[0045] In COMSOL, it is necessary to discretize the selected study area and divide it into fine grids. Then, within each grid cell, the electric field can be approximately expanded using vector basis functions as: where n is the number of expansion basis functions in grid cell e, is the Whitney basis function of the i-th expansion basis, is the electric field expansion coefficient corresponding to .

[0046] After the calculation is completed, select the one-dimensional plot group in the results of COMSOL, select the wavelength as the X-axis variable of the image, and the absorption factor Q abs as the Y-axis variable, then the absorption factor graph can be plotted. Select the two-dimensional plot group, select the X-Y plane as the cross section of the image, and the electric field mode as the variable, then the electric field distribution graph can be plotted.

[0047] Example:

[0048] The above method does not specify the metal material. Here, Ag nanoparticles are taken as an example to illustrate the present invention in detail.

[0049] Step 1: Establish a nanoparticle model in the finite element software COMSOL.

[0050] The Feibelman d parameter of Ag is based on the data calculated by Christensen et al. (Christensen Thomas et al. Quantum Corrections in Nanoplasmonics: Shape, Scale, and Material. [J]. Physical review letters, 2017, 118(15):157402 - 157409.), where 3.81 eV, as Figure 3 .

[0051] Step 2: Set the material parameters.

[0052] Set the relative magnetic permeability of the material to 1, the conductivity to 0 S / m, and the relative permittivity is obtained from the classical Drude model: where ε(ω) is the dielectric constant of silver, ε(∞) is the background dielectric constant of silver obtained from Johnson's experimental data (P.B. Johnson and R.W. Christy. Optical Constants of the Noble Metals [J]. Physical Review B, 1972, 6(12):4370 - 4379.), ω is the angular frequency, ω p is the plasma frequency of the particle, γ is the damping constant of the particle, j is the imaginary unit; and the values that are more in line with the experimental results are selected: ( P A D, Christensen T, Rivera N H, et al. Plasmon - emitter interactions at the nanoscale [J]. Nature Communications, 2020, 11(1):366 - 379), where is the reduced Planck constant.

[0053] Step 3: Set the integration region for calculating the absorption factor

[0054] Step 4: Perform the calculation and export the results

[0055] Add parametric sweep under the research module in COMSOL, and set the variation range of the incident light wavelength to be 200 - 500 nm (with an interval of 50 nm), and the variation range of the radius of Ag nanoparticles to be 5 - 20 nm (with an interval of 5 nm).

[0056] Export the calculated absorption factor diagrams of the classical model as Figure 4 , and the absorption factor diagrams of the modified model as Figure 5 . The electric field distributions of the classical model and the modified model at 220 nm and 280 nm are as Figure 6 .

[0057] From Figure 4 Figure 5 it can be seen that in the classical model, as the particle radius decreases, the resonance absorption peak undergoes a red shift, from 260 nm when the radius is 20 nm to 340 nm when the radius is 5 nm; while the result of the modified model is exactly the opposite. As the particle radius decreases, the resonance absorption peak undergoes a blue shift, from 240 nm when the radius is 20 nm to 210 nm when the radius is 5 nm, which is consistent with the conclusion of Scholl J A et al. (Scholl J A, Ai L K, Dionne J A. Quantum plasmon resonances of individual metallic nanoparticles[J]. Nature, 2012, 483(7390):421 - 427.). And numerically, in the classical model, as the radius decreases, the peak value gradually increases, from 0.77 at 20 nm to 2.62 at 5 nm, while the peak value calculated by the modified model decreases as the radius decreases, from 0.48 at 20 nm to 0.07 at 5 nm, which is also consistent with the conclusion of Scholl J A. In addition, at the same radius, the peak value of the absorption peak calculated by the modified model is significantly smaller than that of the classical model. At 20 nm, it decreases from 0.77 to 0.48, with a decrease of 37.7%; at 5 nm, it decreases from 2.62 to 0.07, with a decrease of 97.3%.

[0058] By comparing the changes in the absorption peak positions of the classical model and the modified model at different radii, it can be seen that for the modified model compared to the classical model, the absorption peak position undergoes an obvious blue shift; and as the particle radius increases, the blue shift amount becomes smaller. At a radius of 5 nm, the modified model is blue - shifted by 120 nm relative to the classical model; while as the radius increases, at a radius of 20 nm, the modified model is only blue - shifted by 20 nm relative to the classical model. This indicates that as the particle radius increases, the modified model and the classical model tend to be consistent, which is in line with the conclusion that there are errors between the classical model and experimental results at the quantum scale, and the errors are larger when the size is smaller.

[0059] From Figure 6 It can be seen from the electric field distribution diagram in Figure 6 that at their respective resonance peaks, the electric field enhancement calculated by the modified model is significantly smaller than that calculated by the classical model. It can be seen that the electric field enhancement of the classical model is mainly concentrated inside the spherical particles. The distribution of the electric field is symmetric along the X-axis and Y-axis, and the electric field inside the particles remains basically unchanged. The electric field distribution at the resonance peak is enhanced along the X-axis direction, and the maximum value is 23.5. In the electric field enhancement region, it gradually decreases from the center of the particle towards the air domain. The closer to the inside of the particle, the stronger the electric field enhancement effect. When it is about the diameter distance outside the particle, the electric field basically returns to the initial field strength.

[0060] The electric field enhancement calculated by the modified model is mainly distributed in the air domain near the spherical particle. Its electric field distribution is also symmetric along the X-axis and Y-axis. The electric field distribution is enhanced along the X-axis direction, and the electric field inside the particle also remains basically unchanged. In the electric field enhancement region, it gradually decreases from the center of the particle towards the air domain. The closer to the air domain near the particle boundary, the stronger the electric field enhancement effect. The enhancement multiple reaches the maximum value at the resonance peak, about 11, and when it is about the radius distance outside the particle, the electric field basically returns to the initial field strength.

Claims

1. A method for calculating the absorption and scattering characteristics of nanoparticles at the quantum scale, characterized in that, it includes the following steps: Step 1, establish a nanoparticle model in the finite element software COMSOL: From the outside to the inside are the perfectly matched layer, the external environment dielectric layer, the weak form computational domain, and the metal particle, where the weak form computational domain is added and the computational domain position is selected in the middle of the external environment dielectric layer and the metal particle region; In the weak form calculation domain, the governing equation is: The weak expression is: where test() is the trial function in COMSOL, ph x 、ph y 、ph z represent the electric field strengths in the X, Y, and Z directions in the weak form computational domain, respectively; V is the volume; k is the propagation constant, and emw.epsilonrxx, emw.epsilonryy, emw.epsilonrzz represent the xx, yy, and zz components of the relative permittivity, respectively; E x 、E y 、E z represent the electric field strengths in the X, Y, and Z directions in the entire space, respectively; r 0 is the correction constant; At the same time, a weak expression is added in the metal particle region, and the weak expression is: -test(ph)*(-ph - dperp / r 0 *(rhob - rhom)) / dperp where ph is the electric field strength in the weak form computational domain; rhob is the surface electric field strength of air, rhom is the surface electric field strength of the metal; dperp is the Feibelman d parameter; Step 2, set the material parameters, including relative magnetic permeability, conductivity, and relative dielectric constant; Step 3, set the integration region for calculating the absorption factor: Set the weak form computational domain and the metal particle region as the integration region for calculating the absorption factor; Step 4, perform calculations and export the results: Add parametric sweep under the study module in COMSOL, set the variation range of the incident light wavelength and the variation range of the radius of the metal nanoparticles, and start the calculation. In the results in COMSOL, select the one-dimensional plot group, and select the wavelength as the X-axis variable of the image and the absorption factor Q as the Y-axis variable. abs , plot the absorption factor graph; select the two-dimensional plot group, select the X-Y plane as the cross-section of the image, and select the electric field mode as the variable to plot the electric field distribution graph; The calculation formula for the surface electric field strength rhob of air is: rhob = up(E x ) * n x + up(E y ) * n y + up(E z ) * n z Among which E x 、E y 、E z respectively represent the electric field strengths in the X, Y, and Z directions in the entire space; up() is the up-nearest estimation expression in COMSOL, n x 、n y 、n z respectively represent the normal vectors in the X, Y, and Z directions; The calculation formula for the surface electric field strength of the metal is: rhom = down(E x ) * n x + down(E y ) * n y + down(E z ) * n z Among them, E x , E y , E z respectively represent the electric field strengths in the X, Y, and Z directions in the entire space; down() is the lower adjacent estimation expression in COMSOL, and n x , n y , n z respectively represent the normal vectors in the X, Y, and Z directions; Absorption factor Q abs The calculation formula is as follows: where C abs is the absorption cross-section and r is the equivalent volume radius of the metal particle.

2. The method for calculating the absorption and scattering characteristics of nanoparticles at the quantum scale according to claim 1, characterized in that, The electric field is approximately expanded using vector basis functions as follows: where n is the number of expansion basis functions in the grid cell e, is the Whitney basis function of the i-th expansion basis, is related to the corresponding electric field expansion coefficient.

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