A Semiclassical Modeling Method and System for Electromagnetic-Quantum Coupling of Nanoantennas

Abstract

This invention discloses a semiclassical modeling method and system for electromagnetic-quantum coupling of nanoantennas, belonging to the field of multiphysics coupling solutions. The modeling method includes: establishing classical electromagnetic model equations and hydrodynamic model equations respectively, and coupling them together through the continuity of the fields to obtain the basic equations of semiclassical hydrodynamics; solving the basic equations of semiclassical hydrodynamics using the finite element method to obtain the electric field and current density; using the electric field and current density for iterative solution to update the coupling terms and obtain the electric field distribution of the nanoparticles, thereby obtaining the optical properties of the nanoantenna; thus achieving a balance between classical electromagnetic field theory and quantum mechanical effects, suitable for the analysis of various nanoantenna structures with moderate computational accuracy requirements; solving the problem that classical electromagnetic field theory cannot well explain and describe small-sized metal nanoantenna structures, and avoiding large-scale calculations using all-quantum methods.

Description

Technical Field

[0001] This invention belongs to the field of multiphysics coupling solution, specifically involving a semiclassical modeling method and system for electromagnetic-quantum coupling of nanoantennas. Background Technology

[0002] The Hydrodynamic Drude Model (HDM) achieves a balance between classical electromagnetic field theory and quantum mechanical effects. It is suitable for analyzing various nanostructures with moderate computational precision requirements and is a classic method for analyzing the nonlocal properties of metallic nanostructures. Its core lies in treating free electrons in metals as a "fluid" and describing their dynamic behavior through hydrodynamic equations. The surface plasmon resonance phenomenon generated by metallic nanostructures can be described based on macroscopic Maxwell's theory, and its dielectric response behavior is mainly based on the Drude-Lorentz bulk material dispersion model, where the dielectric function is only a function of frequency. The Drude-Lorentz bulk material dispersion model provides a good theoretical explanation for large-size materials, but it faces significant challenges in describing small-size metallic nanostructures. These small nanostructures have strong surface interactions and quantum confinement, thus electrons exhibit wave-like quantum properties, modulating their response to electromagnetic waves primarily through nonlocal optical responses and electron overflow. In principle, all-quantum methods, such as density functional theory, can be used to describe the significant quantum effects of small-size metallic nanostructures. However, due to their computational complexity and... (N e The value is proportional to the number of electrons, which obviously makes it impossible to theoretically calculate the optical properties of actual plasmonic devices and cannot handle complex structural systems. Summary of the Invention

[0003] To address the shortcomings of existing technologies, the present invention aims to provide a semiclassical modeling method and system for electromagnetic-quantum coupling of a nanoantenna, thereby solving the problems in the prior art.

[0004] The objective of this invention can be achieved through the following technical solutions:

[0005] A semiclassical modeling method for electromagnetic-quantum coupling of nanoantennas includes the following steps:

[0006] The classical electromagnetic model equations and the fluid dynamics model equations are established separately and coupled together through the continuity of the field to obtain the basic equations of semiclassical fluid dynamics.

[0007] The fundamental equations of semiclassical fluid dynamics are solved using the finite element method to obtain the electric field and current density;

[0008] By iteratively solving the electric field and current density, the coupling terms are updated to obtain the electric field distribution of the nanoparticles, and thus the optical properties of the nanoantenna are obtained.

[0009] Furthermore, the classical electromagnetic model equations are as follows:

[0010]

[0011] Where k0 is the wave number in vacuum, E is the electric field strength, and ε r It is the relative permittivity, μ r ω is the relative permeability, μ is the angular frequency, J is the current density, i is the imaginary unit, and ▽ is the gradient operator.

[0012] Furthermore, the fluid dynamics model equations are as follows:

[0013]

[0014] Where β = (3 / 5) 1 / 2 ν F It is a hydrodynamic parameter describing the strength of electron-electron interactions; ν F It is the Fermi velocity; ε0 is the vacuum conductivity, γ and ω P These are the damping coefficient and the plasma frequency, respectively.

[0015] Furthermore, the steps to solve the fundamental equations of semiclassical fluid dynamics include:

[0016] The tetrahedral discrete nanoantenna is used to decompose the complete structure into a finite number of elements for solution. The edges are defined, vector functions are constructed, and the vector basis functions of the edges are used to replace the solution of the electric field components.

[0017] Weak solution forms for the classical electromagnetic model equations and the fluid dynamics model equations are derived respectively.

[0018] After discretizing the classical electromagnetic model equations and the fluid dynamics model equations respectively, the multiphysics field is solved by bidirectional coupling.

[0019] Furthermore, when deriving the weak solution forms of the classical electromagnetic model equations and the hydrodynamic model equations, the equations are multiplied by the test function and integrated over the computational domain to weaken the derivative terms.

[0020] Furthermore, the specific implementation steps of bidirectional coupling are as follows:

[0021] First, solve for the electric field strength and the effect of the electric field on the flow velocity in the fluid field in the classical electromagnetic model equations.

[0022] Then solve for the current density in the fluid dynamics model equations;

[0023] Finally, the current density is coupled into the classical electromagnetic model equations to achieve bidirectional coupling.

[0024] A semiclassical modeling system for electromagnetic-quantum coupling of nanoantennas, comprising:

[0025] Basic Equation Construction Module: Establish the classical electromagnetic model equations and the fluid dynamics model equations respectively, and couple them together through the continuity of the field to obtain the semi-classical fluid dynamics basic equations;

[0026] Equation solving module: The finite element method is used to solve the basic equations of semiclassical fluid dynamics to obtain the electric field and current density;

[0027] In addition, the iterative update module: uses electric field and current density to iteratively solve and update coupling terms to obtain the electric field distribution of nanoparticles, and then obtains the optical properties of nanoantennas.

[0028] A computer storage medium storing a readable program that, when executed, can perform the aforementioned electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna.

[0029] An electronic device includes: a processor, a memory, a communication interface, and a communication bus, wherein the processor, the memory, and the communication interface communicate with each other through the communication bus;

[0030] The memory is used to store at least one executable instruction, which causes the processor to perform the operation corresponding to the electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna described above.

[0031] A computer program product includes computer instructions that instruct a computing device to perform the operation corresponding to the above-described electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna.

[0032] The beneficial effects of this invention are:

[0033] 1. This invention modifies the Drude model by introducing the quantum properties of electrons, thereby avoiding the inability of classical electromagnetic field theory to adequately explain and describe small-sized metal nanostructures, and avoiding large-scale calculations of the all-quantum method.

[0034] 2. This invention extends the classical electromagnetic equation model, making it highly compatible with traditional electromagnetic field theory, so as to combine it with existing numerical technology tools and quickly realize the analysis of nonlocal characteristics. Attached Figure Description

[0035] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0036] Figure 1 This is a flowchart of the electromagnetic-quantum coupling semiclassical modeling method for the nano-antenna of the present invention;

[0037] Figure 2 This is a structural diagram of the metal nanoparticle antenna of the present invention;

[0038] Figure 3 This is an optical cross-sectional view of the metal nanoparticle antenna of the present invention;

[0039] Figure 4 This is an extinction cross-section diagram of the nanoparticle antenna of this invention under localized and non-localized effects. Detailed Implementation

[0040] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0041] Example 1

[0042] like Figure 1 As shown, a semiclassical modeling method for electromagnetic-quantum coupling of nanoantennas includes the following steps:

[0043] S1. Establish the classical electromagnetic model equations and the fluid dynamics model equations respectively, and couple them together through the continuity of the field to obtain the semi-classical fluid dynamics fundamental equations.

[0044] According to Maxwell's equations and constitutive relations, the wave equation (i.e., the classical electromagnetic model equations) for an arbitrary structured field is:

[0045]

[0046] Where k0 is the wave number in vacuum, E is the electric field strength, and ε r It is the relative permittivity, μ r ε is the relative permeability, ω is the angular frequency, μ is the permeability, J is the current density, i is the imaginary unit, and ▽ is the gradient operator. The right side of the equation represents the nonlocal displacement current. The dielectric constant of metallic materials is represented by the Drude model, i.e., ε D (ω)=ε∞ -iσ D / ε0ω, where ε0 is the Drude conductivity. ε0 is the vacuum conductivity, and γ and ω0 are also relevant. P These are the damping coefficient and the plasma frequency, respectively. γ = 1 / τ, where τ is the relaxation time of the free electron gas. Assume ε... ∞ =1 and ignore interband electron transitions.

[0047] The hydrodynamic model equations are derived from the internal energy of the electron plasma. When the nanostructure size decreases, spatial dispersion and quantum interactions are considered, electron tunneling is disregarded, and electrons are assumed to move only within their own electron cells. Therefore, the Thomas-Fermi dynamics function is chosen as the internal energy functional, resulting in the following nonlocal displacement current transport equation in the hydrodynamic model:

[0048]

[0049] Where β = (3 / 5) 1 / 2 ν F This is a hydrodynamic parameter describing the strength of electron-electron interactions. ν F It is the Fermi velocity; ε0 is the vacuum conductivity, γ and ω P These are the damping coefficient and the plasma frequency, respectively.

[0050] In summary, the coupled equations of the wave equation and the fluid dynamics model equations constitute the fundamental equations of semiclassical fluid dynamics.

[0051] S2, the basic equations of semiclassical fluid dynamics are solved using the finite element method to obtain the electric field and current density;

[0052] The steps to solve the fundamental equations of semiclassical fluid dynamics include:

[0053] S21 employs a tetrahedral discrete nanoantenna, decomposing the complete structure into a finite number of elements for solution, defining edges, constructing vector functions, and using the vector basis functions of the edges to replace the solution of electric field components;

[0054] First, the edges of the tetrahedral element are defined. The interpolation basis function within the tetrahedral element is expressed as follows:

[0055]

[0056] Among them, V e Let e ​​be the volume of the e-th tetrahedron. Let be the coordinates of the four vertices of the e-th tetrahedral element. and These are the expansion coefficients of the interpolation basis functions.

[0057] Then, construct the vector function. The vector basis functions for any edge can then be obtained, that is:

[0058]

[0059] Where i1 and i2 represent the numbers of the first and second nodes on the i-th edge, respectively, which are determined by the definition of an edge. It is the length of the i-th edge.

[0060] Therefore, the electric field components can be expressed as:

[0061]

[0062] in, Find the coefficient of the i-th edge of the e-th tetrahedral element.

[0063] S22, derive the weak solution forms of the equations satisfied by the electric field components and the current components respectively, that is, derive the weak solution forms of the classical electromagnetic model equations and the hydrodynamic model equations respectively.

[0064] For the classical electromagnetic model equations, the derivative terms are weakened by multiplying by the test function and integrating over the computational domain; the fluid dynamics equations are weakened using the same method, as follows:

[0065] Before deriving the equations, the boundary conditions must be addressed. Generally, standard boundary conditions are sufficient for solving classical electric field model equations. However, considering quantum nonlocal effects, additional boundary conditions need to be added for the current. According to the literature, the boundary conditions for the current are:

[0066]

[0067] Where n is the normal vector at the boundary. It is the boundary surface of an object.

[0068] 1) First, derive the weak solution form of equation (1) of the classical electromagnetic model, that is, by multiplying the equation by the test function and integrating it. This is the test function:

[0069]

[0070] For the first term, use the vector identity. And by simplifying with the Gaussian divergence theorem, we get:

[0071]

[0072] Substituting equation (8) and the edge vector function into equation (7), we can obtain the weak solution form of equation (1):

[0073]

[0074] 2) Next, derive the weak solution form of the fluid dynamics equation (2). This is the test function. Let ▽·J = u, and use the vector identity... By integration, we can obtain:

[0075]

[0076] Using the Gaussian divergence theorem and boundary conditions, we can obtain that the result of the first term on the right side of equation (10) is zero.

[0077] Multiplying the test function by equation (2) and integrating, then substituting equation (10) into the equation, we obtain the weak solution form as follows:

[0078]

[0079] After completing the above steps, weak solutions to the classical electromagnetic model equations and the fluid dynamics model equations can be obtained.

[0080] S23, after discretizing the classical electromagnetic model equations and the fluid dynamics model equations respectively, performs bidirectional coupling solution for the multiphysics field;

[0081] Two-way coupling is achieved through electric field strength E and current density J. The specific implementation steps are as follows: First, solve the electric field strength and the influence of the electric field on the flow velocity in the fluid field in the classical electromagnetic model equations. Then, solve the current density in the fluid dynamics model equations, since the current density also affects the electric field. Finally, couple the current density into the wave equation (classical electromagnetic model equations) to achieve two-way coupling.

[0082] S3 uses the electric field and current density obtained from S2 to iteratively solve and update the coupling terms to obtain the electric field distribution of the nanoparticles, and then obtain the optical properties of the nanoantenna.

[0083] This embodiment studies the optical properties of the cross-sectional spectrum of a nanoantenna, typically referring to physical quantities such as the scattering cross section, absorption cross section, and extinction cross section. The extinction cross section represents the total amount of radiation scattered and absorbed by an object when an electromagnetic wave is incident on it; it is the sum of the scattering and absorption cross sections.

[0084] σ ext =σ scat +σ abs (12)

[0085] in:

[0086]

[0087] In the formula, σ ext ,σ scat,σ abs These are the extinction cross section, scattering cross section, and absorption cross section, respectively. scat ,P abs These are the total scattered power and the total absorbed power, respectively. S is the Poynting vector, n is the normal vector, and E is the total scattered power and the total absorbed power. scat (r,ω),H scat (r, ω) represent the scattering electric field and the scattering magnetic field, respectively, which can be obtained by subtracting the incident field E from the total field E, H. inc H inc H was obtained. * It is the conjugate of the magnetic field.

[0088] Finally, the nonlocal effect of the nanoantenna can be observed based on the extinction cross section.

[0089] Based on a similar inventive concept, this embodiment of the invention also provides a computer storage medium storing a readable program that, when the program is run, can execute the above-described electromagnetic-quantum coupling semiclassical modeling method for nanoantennas.

[0090] Based on a similar inventive concept, this invention provides an electronic device, including: a processor, a memory, a communication interface, and a communication bus, wherein the processor, the memory, and the communication interface communicate with each other through the communication bus;

[0091] The memory is used to store at least one executable instruction, which causes the processor to perform the operation corresponding to the above-described electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna.

[0092] Based on a similar inventive concept, embodiments of the present invention also provide a computer program product, including computer instructions, which instruct a computing device to perform operations corresponding to the above-described electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna.

[0093] Example 2

[0094] In this embodiment, the optical cross-section of a metal nanoparticle antenna is analyzed using the method described in Example 1; as follows: Figure 2 As shown, a simulation of a gold nanosphere with a radius of 2 nm was performed. The dielectric constant of gold was fitted using the Drude model, where the parameters of the Drude model are as follows:

[0095] ω P =8.812eV,γ=0.0752eV,β=(0.6)ν F ,ν F =1.07×10 6 ms -1The surrounding medium is air. A tetrahedral mesh is used, with PML absorbing boundary conditions applied around the perimeter. The incident wave is a plane wave polarized in the x-direction. The electromagnetic-quantum coupling semiclassical modeling method for the nanoantenna is implemented according to the steps described in Example 1.

[0096] Figure 3 The optical cross-section results of the gold nanoparticle antenna calculated by the semi-classical modeling method are presented. Figure 3 As can be seen from this, the extinction cross section is the sum of the scattering cross section and the absorption cross section.

[0097] Figure 4 The extinction cross-sections of gold nanoparticle antennas under local and nonlocal interactions are presented. Figure 4 It can be seen that, compared with the traditional local resonance peak, the nonlocal characteristics have a significant blue shift.

[0098] Example 3

[0099] Based on the semi-classical modeling method for electromagnetic-quantum coupling of nanoantennas mentioned in Example 1, this example proposes a semi-classical modeling system for electromagnetic-quantum coupling of nanoantennas, specifically including:

[0100] Basic Equation Construction Module: Establish the classical electromagnetic model equations and the fluid dynamics model equations respectively, and couple them together through the continuity of the field to obtain the semi-classical fluid dynamics basic equations;

[0101] Equation solving module: The finite element method is used to solve the basic equations of semiclassical fluid dynamics to obtain the electric field and current density;

[0102] In addition, the iterative update module: uses electric field and current density to iteratively solve and update coupling terms to obtain the electric field distribution of nanoparticles, and then obtains the optical properties of nanoantennas.

[0103] The methods of the present invention can be implemented in hardware, firmware, or as software or computer code that can be stored in a recording medium (such as a CD-ROM, RAM, floppy disk, hard disk, or magneto-optical disk), or as computer code originally stored on a remote recording medium or a non-transitory machine-readable medium and subsequently stored on a local recording medium, downloaded via a network. Thus, the methods described herein can be processed by software stored on a recording medium using a general-purpose computer, a dedicated processor, or programmable or dedicated hardware (such as an ASIC or FPGA). It is understood that the computer, processor, microprocessor controller, or programmable hardware includes storage components (e.g., RAM, ROM, flash memory, etc.) capable of storing or receiving software or computer code that, when accessed and executed by the computer, processor, or hardware, implements the methods described herein. Furthermore, when a general-purpose computer accesses the code used to implement the methods shown herein, the execution of the code transforms the general-purpose computer into a dedicated computer for performing the methods shown herein.

[0104] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention.

Claims

1. A semi-classical modeling method for electromagnetic-quantum coupling of nanoantennas, characterized in that, Includes the following steps: The classical electromagnetic model equations and the fluid dynamics model equations are established separately and coupled together through the continuity of the field to obtain the basic equations of semiclassical fluid dynamics. The fundamental equations of semiclassical fluid dynamics are solved using the finite element method to obtain the electric field and current density; By iteratively solving the electric field and current density, the coupling terms are updated to obtain the electric field distribution of the nanoparticles, and thus the optical properties of the nanoantenna are obtained.

2. The semi-classical modeling method for electromagnetic-quantum coupling of nanoantennas according to claim 1, characterized in that, The classical electromagnetic model equations are: Where k0 is the wave number in vacuum, E is the electric field strength, and ε r It is the relative permittivity, μ r ω is the relative permeability, μ is the angular frequency, J is the current density, i is the imaginary unit, and ▽ is the gradient operator.

3. The semi-classical modeling method for electromagnetic-quantum coupling of nanoantennas according to claim 2, characterized in that, The fluid dynamics model equations are as follows: Where β = (3 / 5) 1 / 2 ν F It is a hydrodynamic parameter describing the strength of electron-electron interactions; ν F It is the Fermi velocity; ε0 is the vacuum conductivity, γ and ω P These are the damping coefficient and the plasma frequency, respectively.

4. The semi-classical modeling method for electromagnetic-quantum coupling of nanoantennas according to claim 1, characterized in that, The steps to solve the fundamental equations of semiclassical fluid dynamics include: The tetrahedral discrete nanoantenna is used to decompose the complete structure into a finite number of elements for solution. The edges are defined, vector functions are constructed, and the vector basis functions of the edges are used to replace the solution of the electric field components. Weak solution forms for the classical electromagnetic model equations and the fluid dynamics model equations are derived respectively. After discretizing the classical electromagnetic model equations and the fluid dynamics model equations respectively, the multiphysics field is solved by bidirectional coupling.

5. The semi-classical modeling method for electromagnetic-quantum coupling of nanoantennas according to claim 4, characterized in that, When deriving the weak solution forms of the classical electromagnetic model equations and the hydrodynamic model equations, the equations are multiplied by the test function and integrated over the computational domain to weaken the derivative terms.

6. The semi-classical modeling method for electromagnetic-quantum coupling of nanoantennas according to claim 4, characterized in that, The specific implementation steps of bidirectional coupling are as follows: First, solve for the electric field strength and the effect of the electric field on the flow velocity in the fluid field in the classical electromagnetic model equations. Then solve for the current density in the fluid dynamics model equations; Finally, the current density is coupled into the classical electromagnetic model equations to achieve bidirectional coupling.

7. A semi-classical modeling system for electromagnetic-quantum coupling of nanoantennas, characterized in that, include: Basic Equation Construction Module: Establish the classical electromagnetic model equations and the fluid dynamics model equations respectively, and couple them together through the continuity of the field to obtain the semi-classical fluid dynamics basic equations; Equation solving module: The finite element method is used to solve the basic equations of semiclassical fluid dynamics to obtain the electric field and current density; In addition, the iterative update module: uses electric field and current density to iteratively solve and update coupling terms to obtain the electric field distribution of nanoparticles, and then obtains the optical properties of nanoantennas.

8. A computer storage medium storing a readable program, characterized in that, When the program runs, it can execute the electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna as described in any one of claims 1-6.

9. An electronic device, characterized in that, include: The processor, memory, communication interface, and communication bus are provided, wherein the processor, memory, and communication interface communicate with each other via the communication bus. The memory is used to store at least one executable instruction that causes the processor to perform the operation corresponding to the electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna as described in any one of claims 1-6.

10. A computer program product comprising computer instructions, characterized in that, The computer instructions instruct the computing device to perform the corresponding operation of the electromagnetic-quantum coupling semiclassical modeling method for a nanoantenna as described in any one of claims 1-6.

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