Electromagnetic-quantum coupling semi-classical modeling method and system for nano antenna

Abstract

The invention discloses an electromagnetic-quantum coupling semi-classical modeling method and system for a nano antenna, and belongs to the field of multi-physics field coupling solution. The modeling method comprises the following steps: respectively establishing a classic electromagnetic model equation and a fluid dynamic model equation, and simultaneously obtaining a semi-classic fluid dynamic basic equation through field continuity coupling; solving the semi-classical hydrodynamic basic equation by adopting a finite element method to obtain an electric field and current density; performing iterative solution by using the electric field and the current density, updating the coupling term, obtaining the electric field distribution of the nano particles, and further obtaining the optical characteristics of the nano antenna; therefore, the balance between the classical electromagnetic field theory and the quantum mechanical effect is realized, and the method is suitable for analyzing various nano antenna structures with medium calculation precision requirements; the problem that the classical electromagnetic field theory cannot well explain and describe the small-size metal nano antenna structure is solved, and large-scale calculation using a full-quantum method is avoided.

Description

Technical Field

[0001] The present invention belongs to the field of multi-physical field coupling solution, and specifically relates to an electromagnetic-quantum coupling semiclassical modeling method and system for nanoantennas. Background Art

[0002] The hydrodynamic Drude Model (HDM) achieves a balance between classical electromagnetic field theory and quantum mechanical effects. It is suitable for the analysis of various nanostructures with medium computational accuracy requirements. It is a classic method for analyzing the non-local characteristics of metal nanostructures. Its core is to regard the free electrons in the metal as "fluids" and describe their dynamic behavior through hydrodynamic equations. The surface plasmon phenomenon generated by metal nanostructures can be described based on the macroscopic Maxwell theory. Its dielectric response behavior is mainly based on the Drude-Lorentz bulk material dispersion model, and the dielectric function is only a function of frequency. The Drude-Lorentz bulk material dispersion model can give a good theoretical explanation when describing large-scale materials, but it encounters great challenges when describing small-scale metal nanostructures. This small-scale nanostructure has strong surface interactions and quantum confinement, so electrons exhibit quantum characteristics of waves, and their response behavior to electromagnetic waves is mainly modulated by non-local optical responses and electron overflows. In principle, full quantum methods, such as density functional theory, can be used to describe the significant quantum effects of small-scale metal nanostructures. However, due to its computational complexity and (N e This obviously makes it impossible to perform theoretical calculations on the optical properties of actual plasmon devices and cannot handle complex structural systems. Summary of the invention

[0003] In view of the deficiencies in the prior art, the purpose of the present invention is to provide an electromagnetic-quantum coupling semiclassical modeling method and system for a nanoantenna, which solves the problems in the prior art.

[0004] The purpose of the present invention can be achieved by the following technical solutions:

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

[0006] The classical electromagnetic model equations and the fluid dynamics model equations are established respectively, and coupled through the continuity of the field, and the basic equations of semi-classical fluid dynamics are obtained together;

[0007] The finite element method is used to solve the basic equations of semiclassical fluid dynamics to obtain the electric field and current density;

[0008] The electric field and current density are iteratively solved to update the coupling terms, obtain the electric field distribution of the nanoparticles, and then obtain the optical properties of the nanoantenna.

[0009] Furthermore, the classical electromagnetic model equation is:

[0010]

[0011] Where k0 is the wave number in vacuum, E is the electric field strength, and ε r is the relative dielectric constant, μ r 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.

[0012] Furthermore, the fluid dynamics model equation is:

[0013]

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

[0015] Further, the steps to solve the basic equations of semiclassical fluid dynamics include:

[0016] The nanoantenna is discretely divided into tetrahedrons, the complete structure is decomposed into a finite number of units for solution, edges are defined, vector functions are constructed, and the vector basis functions of the edges are used to solve the electric field components.

[0017] The weak solutions of 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 multi-physical fields are solved by bidirectional coupling.

[0019] Furthermore, when deriving the weak solution forms of the classical electromagnetic model equations and the fluid dynamics 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 the bidirectional coupling are:

[0021] First, solve the electric field intensity in the classical electromagnetic model equation and the flow velocity in the fluid field affected by the electric field;

[0022] The fluid dynamics model equations are then solved for current density;

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

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

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

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

[0027] And, iterative update module: use the electric field and current density to iteratively solve, update the coupling terms, obtain the electric field distribution of the nanoparticles, and then obtain the optical properties of the nanoantenna.

[0028] A computer storage medium stores a readable program, which can execute the above-mentioned electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna when the program is running.

[0029] An electronic device, comprising: 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, and the executable instruction enables the processor to execute the above-mentioned electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna and corresponding operations.

[0031] A computer program product includes computer instructions, wherein the computer instructions instruct a computing device to execute the above-mentioned electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna and corresponding operations.

[0032] Beneficial effects of the present invention:

[0033] 1. The present invention introduces the quantum properties of electrons and modifies the Drude model to avoid the problem that the classical electromagnetic field theory cannot well explain and describe small-sized metal nanostructures, and to avoid large-scale calculations using the full quantum method.

[0034] 2. The present invention is expanded through the classical electromagnetic equation model and is highly compatible with traditional electromagnetic field theory to achieve integration with existing numerical technology tools and quickly realize non-local characteristic analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, for ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0036] Figure 1 It is a flow chart of the electromagnetic-quantum coupling semiclassical modeling method of the nanoantenna of the present invention;

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

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

[0039] Figure 4 It is the extinction cross-section diagram of the nanoparticle antenna of the present invention under the local effect and the non-local effect. DETAILED DESCRIPTION

[0040] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0041] Example 1

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

[0043] S1, respectively establish the classical electromagnetic model equations and the fluid dynamics model equations, and couple them through the continuity of the field to obtain the basic equations of semi-classical fluid dynamics;

[0044] According to Maxwell's equations and constitutive relations, the wave equation of any structural field (i.e., the classical electromagnetic model equation) is:

[0045]

[0046] Where k0 is the wave number in vacuum, E is the electric field strength, and ε r is the relative dielectric constant, μ r 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 is the nonlocal displacement current. The dielectric constant of the metal material adopts the Drude model, that is, ε D (ω)=ε∞ -iσ D / ε0ω, where is the Drude conductivity. ε0 is the vacuum conductivity, γ and ω P are the damping coefficient and plasma frequency respectively. γ=1 / τ, τ is the relaxation time of the free electron gas. Assume ε ∞ =1 and interband electronic transitions are ignored.

[0047] The equation of the fluid dynamics model is given based on the internal energy of the electron plasma. When the size of the nanostructure is reduced, spatial dispersion and quantum effects are considered, electron tunneling is not considered, and only the electrons are considered to move inside the electrons. Therefore, the Thomas-Fermi dynamics function is selected as the internal energy functional function, and the non-local displacement current transmission equation in the fluid dynamics model is obtained as follows:

[0048]

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

[0050] To sum up, the coupled equations of wave equation and fluid dynamics model equation are the basic equations of semiclassical fluid dynamics.

[0051] S2, the finite element method is used to solve the basic equations of semiclassical fluid dynamics to obtain the electric field and current density;

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

[0053] S21, using tetrahedron discrete dissection nanoantenna, decomposing the complete structure into a finite number of units for solution, defining edges, constructing vector functions, and using the vector basis functions of the edges to solve the electric field components instead;

[0054] First, define the edges of the tetrahedral element. is the interpolation basis function within the tetrahedral unit, and the calculation expression is:

[0055]

[0056] Among them, V e is the volume of the e-th tetrahedron, are the coordinates of the four vertices of the e-th tetrahedral unit, and are the expansion coefficients of the interpolation basis function.

[0057] Then, construct the vector function The vector basis function of any edge can be obtained, namely:

[0058]

[0059] Among them, i1 and i2 represent the numbers of the first and second nodes on the i-th edge, respectively, which are determined according to the edge definition. is the length of the i-th edge.

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

[0061]

[0062] in, Find the coefficient for the band on the ith edge of the eth tetrahedral element.

[0063] S22, derive the weak solution forms of the equations satisfied by the electric field component and the current component, that is, derive the weak solution forms of the classical electromagnetic model equations and the fluid dynamics model equations respectively;

[0064] For the classical electromagnetic model equation, multiply the test function and integrate it over the computational domain to weaken the derivative term; for the fluid dynamics equation, the same method is used to weaken it, as follows:

[0065] Before deriving the equation, we need to deal with the boundary conditions. Generally, the standard boundary conditions are sufficient to solve the classical electric field model equations. However, considering the quantum non-local effect, we need to add additional boundary conditions to the current. According to the literature, the boundary conditions for the current are:

[0066]

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

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

[0069]

[0070] For the first term, use the vector identity And the Gaussian divergence theorem can be simplified to:

[0071]

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

[0073]

[0074] 2) Then derive the weak solution form of the fluid dynamics equation (2), is the test function. Let ▽·J=u, using the vector identity And the integral can be obtained:

[0075]

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

[0077] Multiplying the test function in equation (2) and integrating it, and then substituting (10) into equation (10), we can get the weak solution in the form of:

[0078]

[0079] After completing the above steps, the weak solution forms of 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, a bidirectional coupling solution is performed on the multi-physics field;

[0081] Bidirectional 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 in the classical electromagnetic model equation, the influence of the electric field on the flow velocity in the fluid field, etc., then solve the current density in the fluid dynamics model equation, and the current density has an impact on the electric field, and finally couple the current density to the wave equation (classical electromagnetic model equation) to achieve bidirectional coupling.

[0082] S3, using the electric field and current density obtained in S2 to perform iterative solution, update the coupling terms, obtain the electric field distribution of the nanoparticles, and then obtain the optical properties of the nanoantenna.

[0083] This example studies the optical properties of the cross-sectional spectrum of the nanoantenna, which usually refers 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, which is the sum of the scattering cross section and the absorption cross section:

[0084] σ ext =σ scat +σ abs (12)

[0085] in:

[0086]

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

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

[0089] Based on similar inventive concepts, an embodiment of the present invention further provides a computer storage medium storing a readable program, which can execute the above-mentioned electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna when the program is running.

[0090] Based on similar inventive concepts, an embodiment of the present invention provides an electronic device, comprising: 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, and the executable instruction enables the processor to perform operations corresponding to the above-mentioned electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna.

[0092] Based on similar inventive concepts, an embodiment of the present invention further provides a computer program product, including computer instructions, which instruct a computing device to execute operations corresponding to the above-mentioned electromagnetic-quantum coupling semiclassical modeling method of 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 Embodiment 1; Figure 2 As shown in the figure, a gold nanosphere with a radius of 2 nm is simulated, and the dielectric constant of gold is fitted using the Drude model, where the Drude model parameters are

[0095] ω P =8.812eV,γ=0.0752eV,β=(0.6)ν F ,ν F =1.07×10 6 ms -1, the surrounding medium is air. Tetrahedral meshing is adopted, PML absorbing boundary conditions are adopted on all sides, and the incident wave is a plane wave polarized in the x direction. According to the steps described in Example 1, the electromagnetic-quantum coupling semiclassical modeling method of the nanoantenna is implemented.

[0096] Figure 3 The optical cross section results of the gold nanoparticle antenna calculated by the semiclassical modeling method are given. Figure 3 It can be seen that the extinction cross section is the sum of the scattering cross section and the absorption cross section.

[0097] Figure 4 The extinction cross section results of gold nanoparticle antennas under local and non-local effects are given. Figure 4 It can be seen that the non-local characteristics have an obvious blue shift compared with the traditional localized resonance peak.

[0098] Example 3

[0099] Based on the electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna mentioned in Example 1, in this embodiment, an electromagnetic-quantum coupling semiclassical modeling system of a nanoantenna is proposed, which specifically includes:

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

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

[0102] And, iterative update module: use the electric field and current density to iteratively solve, update the coupling terms, obtain the electric field distribution of the nanoparticles, and then obtain the optical properties of the nanoantenna.

[0103] The method of the present invention may be implemented in hardware, firmware, or as software or computer code that may be stored in a recording medium (such as a CDROM, RAM, floppy disk, hard disk, or magneto-optical disk), or as computer code that is originally stored in a remote recording medium or a non-temporary machine-readable medium downloaded over a network and will be stored in a local recording medium, so that the method described herein may be stored in such software processing 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 a computer, processor, microprocessor controller, or programmable hardware includes a storage component (e.g., RAM, ROM, flash memory, etc.) that can store or receive software or computer code, and when the software or computer code is accessed and executed by a computer, processor, or hardware, the method described herein is implemented. In addition, when a general-purpose computer accesses the code for implementing the method shown herein, the execution of the code converts the general-purpose computer into a dedicated computer for executing the method shown herein.

[0104] The above shows and describes 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, and the above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, and these changes and improvements all fall within the scope of the present invention to be protected.

Claims

1. A semiclassical modeling method for electromagnetic-quantum coupling of nanoantennas, characterized in that: The following steps are involved: The classical electromagnetic model equations and the fluid dynamics model equations are established respectively, and coupled through the continuity of the field, and the basic equations of semi-classical fluid dynamics are obtained together; The finite element method is used to solve the basic equations of semiclassical fluid dynamics to obtain the electric field and current density; The electric field and current density are iteratively solved to update the coupling terms, obtain the electric field distribution of the nanoparticles, and then obtain the optical properties of the nanoantenna.

2. The electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna according to claim 1, characterized in that: The classical electromagnetic model equation is: Where k0 is the wave number in vacuum, E is the electric field strength, and ε r is the relative dielectric constant, μ r 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.

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

4. The electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna according to claim 1, characterized in that: The steps to solve the basic equations of semiclassical fluid dynamics include: The nanoantenna is discretely divided into tetrahedrons, the complete structure is decomposed into a finite number of units for solution, edges are defined, vector functions are constructed, and the vector basis functions of the edges are used to solve the electric field components. The weak solutions of 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 multi-physical fields are solved by bidirectional coupling.

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

6. The electromagnetic-quantum coupling semiclassical modeling method of a nanoantenna according to claim 4, characterized in that: The specific implementation steps of bidirectional coupling are: First, solve the electric field intensity in the classical electromagnetic model equation and the flow velocity in the fluid field affected by the electric field; The fluid dynamics model equations are then solved for current density; Finally, the current density is coupled into the classical electromagnetic model equation to achieve bidirectional coupling.

7. An electromagnetic-quantum coupling semiclassical modeling system for nanoantennas, characterized in that: include: Basic equation building module: Establish the classical electromagnetic model equations and fluid dynamics model equations respectively, and couple them through field continuity to obtain the basic equations of semi-classical fluid dynamics; Equation solving module: uses the finite element method to solve the basic equations of semiclassical fluid dynamics to obtain the electric field and current density; And, iterative update module: use the electric field and current density to iteratively solve, update the coupling terms, obtain the electric field distribution of the nanoparticles, and then obtain the optical properties of the nanoantenna.

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

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

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