WO2023115815A1 - Method for solving phonon heat transport at interface, and storage medium - Google Patents

Abstract

The present application relates to a method for solving phonon thermal transport at an interface, and a storage medium. The method comprises: initializing set parameters and system parameters; acquiring a phonon deviation intensity on the basis of the system parameters; iteratively updating the phonon deviation intensity on the basis of a linear Boltzmann equation and a phonon interface condition, and obtaining a corresponding temperature, pseudo-temperature and heat flow on the basis of any phonon deviation intensity; acquiring the difference between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference to be a relative error; and comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, outputting the temperature, pseudo-temperature and heat flow corresponding to the current phonon deviation intensity.

Description

A method and storage medium for solving phonon heat transport at an interface

Cross References to Related Applications

This application claims the priority of the Chinese patent application submitted to the China Patent Office on December 24, 2021, with the application number 202111594201.5, and the application name is "A method and storage medium for solving phonon heat transport at the interface", all of which The contents are incorporated by reference in this application.

technical field

The present application relates to the technical field of heat transfer, in particular to a method and a storage medium for solving the heat transport of phonons at an interface.

Background technique

The heat dissipation problem of electronic devices has become a major obstacle restricting its further development. The research on the heat transport mechanism of phonons in semiconductors can provide effective guidance for the heat dissipation optimization design of electronic devices. The research methods mainly include theoretical calculation, experimental research and numerical simulation. As a numerical method based on directly solving the phonon Boltzmann equation, the discrete coordinate method is an important numerical tool for studying phonon heat transport in semiconductors. The discrete coordinate method has obvious advantages for the simulation of mesoscopic scale systems with simple geometric shapes. Its algorithm is simple to implement and the simulation accuracy is high. However, the numerical algorithm framework of the phonon discrete coordinate method is still not perfect.

In the prior art, the phonon discrete coordinate method considering the actual dispersion relationship is mainly divided into two categories: numerical algorithms based on linearized equations under small temperature differences and numerical algorithms based on nonlinear equations under arbitrary temperature differences. The two types of numerical algorithms can further consider steady-state and transient situations, corresponding to space-domain and time-space domain algorithms. Numerical algorithms based on linear Boltzmann equations are simpler and more efficient than those based on nonlinear Boltzmann equations. Using the discrete coordinate method, it is possible to simulate the heat transport process of phonons in a system without an interface or with an interface. Unlike the former, the latter needs to consider the information exchange mechanism of phonons at the interface in the materials on both sides of the interface. The process is more complicated.

The inventor realizes that at present, for the phonon discrete coordinate method based on the linearization equation, the space-domain and space-time domain algorithm framework of the discrete coordinate method without considering the interface system, and the space-domain algorithm framework of the discrete coordinate method considering the interface system have been established , but the existing time-space domain algorithm framework of the discrete coordinate method considering the interface system still needs to be further improved to meet the increasingly urgent numerical simulation needs of the study of phonon heat transport in semiconductors.

Contents of the invention

In order to achieve the above object, the application proposes the first technical solution:

A method for solving phonon heat transport at an interface, the method comprising the following steps: step S1, initializing setting parameters and system parameters; step S2, obtaining phonon deviation strength based on system parameters; step S3, based on linear Bohr Zeman equation and phonon interface conditions, update the phonon deviation strength iteratively, and obtain the corresponding temperature, pseudo temperature and heat flow based on any phonon deviation strength; step S4, obtain the temperature corresponding to the current phonon deviation strength and the previous phonon deviation strength The temperature difference corresponding to the deviation intensity, and define the difference as a relative error; step S5, compare the relative error with the preset convergence error, if the relative error is not greater than the convergence error, then output the current phonon deviation intensity corresponding to temperature, pseudo temperature and heat flow.

In one of the embodiments, the method further includes: step S6, obtaining the calculation time length for iteratively updating the phonon deviation strength, and defining the calculation time length as the first calculation time length; step S7, if the first calculation time length is less than the preset total calculation time duration, then continue to execute steps S3, S4, S5, S6 and S7; otherwise, output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity.

In one of the embodiments, the setting parameters include: space step size, system scale, time step size, total calculation time and convergence error; system parameters include: temperature and pseudo temperature; phonon deviation strength includes: based on phonon strength acquisition The obtained first bias strength and the obtained second bias strength are acquired based on the phonon pseudo-balance strength.

In one embodiment, step S2 specifically includes: step S21, the initial value of the set temperature is T ₀ , based on the initial value of the temperature, the obtained phonon intensity is:

In the formula, I _{ω, p} is the phonon intensity, v _g is the phonon group velocity,

is the reduced Planck constant, ω is the phonon angular frequency, p is the phonon branch, f _{ω, p} is the phonon distribution, D is the phonon density of state, D(ω, p) means that the phonon branch is p and The phonon density of state at which the phonon angular frequency is ω; step S22, based on the phonon intensity, the obtained phonon reference equilibrium intensity is:

In the formula,

is the phonon reference equilibrium strength,

For considering the Bose-Einstein distribution of the constant type reference temperature T ^ref ; step S23, based on the phonon intensity and the phonon reference equilibrium intensity, obtain the first deviation intensity:

In the formula, Ψ _{ω, p} is the first deviation intensity, I _{ω, p} is the phonon intensity,

is the phonon reference equilibrium strength; step S24, setting the initial value of the pseudo-temperature equal to the initial value of the equilibrium value is T ₀ , based on the initial value of the pseudo-temperature, the pseudo-equilibrium strength of the obtained phonon is:

In the formula,

is the phonon pseudo-equilibrium intensity,

Is to consider the Bose-Einstein distribution of the initial pseudo-temperature T ^pse ; step S25, based on the phonon intensity and the pseudo-equilibrium intensity of the phonon, obtain the second deviation intensity:

In the formula,

is the second bias strength.

In one embodiment, the linear Boltzmann equation is:

In the formula, Ψ _{ω, p} is the first deviation intensity,

is the second deviation strength, τ _{ω, p} is the phonon relaxation time, ω is the phonon angular frequency, p is the phonon branch, v _g is the phonon group velocity, v _g (ω, p) is the phonon branch as The phonon group velocity of p and phonon angular frequency ω; the interface condition is:

In the formula, A and B represent two materials respectively, φ is the angle between the interface normal vector and the group velocity, the positive direction of the interface normal is from material A to material B, α _AB (ω) and α _AB (ω) are respectively Spectrum interface penetration coefficient from material A to material B and material B to material A, Ψ _{B, ω, p} is the first deviation intensity of material B, Ψ _{A, ω, p} is the first deviation intensity of material A.

In one of the embodiments, obtaining the corresponding temperature, pseudo temperature and heat flow based on any phonon deviation intensity specifically includes: step S31, based on the basic definition of phonon deviation intensity and temperature, obtaining the linear expression of temperature:

In the formula,

To consider the Bose-Einstein distribution at temperature T, T ^ref is a constant reference temperature, θ is a polar coordinate,

is the azimuth angle, ω _{max, p} represents the maximum phonon angular frequency of the phonon branch p, v _g is the phonon group velocity, Ψ _{ω, p} is the first deviation strength,

is the spectral volume heat capacity, that is, the volume heat capacity of each phonon branch and unit phonon angle frequency range;

Among them, the basic definition of temperature is:

Step S32, based on the basic definition of phonon deviation strength and pseudo temperature, obtain the linear expression of pseudo temperature:

In the formula, T ^pse is the pseudo temperature, τ _{ω, p} is the phonon relaxation time;

Among them, the basic definition of pseudo temperature is:

Step S33, based on the phonon deviation intensity, obtain the linear expression of the heat flow:

In the formula, q is heat flow.

In one of the embodiments, step S5 specifically includes: step S51, obtaining the relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:

In the formula, ε represents the relative error, n represents the spatial unit index; N represents the total number of spatial units; i represents the iteration number index, and T is the temperature;

Step S52, if the relative error is not greater than the convergence error, then output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity.

In one of the embodiments, step S5 specifically includes:

Step S53. Obtain the difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and define the difference as a relative error:

In the formula, ε is the relative error; n is the spatial unit index; N is the total number of spatial units; i is the iteration number index, and q is the heat flow;

Step S54 , comparing the relative error with the preset convergence error, and if the relative error is not greater than the convergence error, then output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity.

In one of the embodiments, the corresponding temperature, pseudo temperature and heat flow are obtained based on any phonon deviation intensity, including:

Obtain the temperature corresponding to any phonon deviation intensity based on the ratio of any phonon deviation intensity to the basic definition of temperature;

Based on the ratio of the basic definition of any phonon deviation intensity to the pseudo temperature, the pseudo temperature corresponding to any phonon deviation intensity is obtained;

Integral summation is performed based on any phonon deviation intensity to obtain the heat flow corresponding to any phonon deviation intensity.

In order to achieve the above object, the application proposes a second technical solution:

One or more non-volatile computer-readable storage media storing computer-readable instructions, when the computer-readable instructions are executed by one or more processors, one or more processors are made to execute the method provided by any one of the above-mentioned embodiments Steps in a method to solve for phonon heat transport at an interface.

The details of one or more embodiments of the application are set forth in the accompanying drawings and the description below. Other features and advantages of the application will be apparent from the description, drawings, and claims.

Description of drawings

In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings that need to be used in the description of the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present application. For those skilled in the art, other drawings can also be obtained based on these drawings without creative effort.

FIG. 1 is an internal structural diagram of a computer device provided according to one or more embodiments of the present application;

Fig. 2 is a flow chart of a method for solving phonon heat transport at an interface according to one or more embodiments of the present application;

Fig. 3 is a flow chart of a method for solving phonon heat transport at an interface according to one or more embodiments of the present application;

FIG. 4 is a schematic structural view of a double-layer film provided according to one or more embodiments of the present application;

Fig. 5 is a temperature evolution experiment result diagram of a double-layer film provided according to one or more embodiments of the present application;

Fig. 6 is a graph showing experimental results of heat flow evolution of a double-layer thin film according to one or more embodiments of the present application.

Detailed ways

In order to make the purpose, technical solutions and advantages of the application clearer, the technical solutions in the embodiments of the application will be clearly and completely described below in conjunction with the drawings in the embodiments of the application. Obviously, the described embodiments are only Some embodiments of this application are not all embodiments. Based on the embodiments in this application, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the scope of protection of this application.

Embodiment one:

This embodiment provides a method for solving the heat transport of phonons at the interface, and the method can be applied to a computer device. The computer device can be a server, and its internal structure diagram can be shown in FIG. 1 . The computer device includes a processor, memory, network interface and database connected by a system bus. Wherein, the processor of the computer device is used to provide calculation and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, computer readable instructions and a database. The internal memory provides an environment for the execution of the operating system and computer readable instructions in the non-volatile storage medium. The database of the computer equipment is used to store data such as setting parameters and system parameters. The network interface of the computer device is used to communicate with an external terminal via a network connection. The computer readable instructions, when executed by the processor, implement the method for solving the phonon heat transport at the interface. Wherein, the server may be implemented by an independent server or a server cluster composed of multiple servers.

Referring to FIG. 2 , FIG. 2 is a flow chart of the method in Embodiment 1.

The method of this embodiment includes the following steps:

Step S1, initialize setting parameters; wherein, the setting parameters include space step, system scale, time step, total calculation time and convergence error; system scale is the system's spatial scale, such as: system length, width and height. Because it only involves one-dimensional space, that is, this application only considers the change of system parameters in the thickness direction of the experimental material, that is, if only the change of the system in the length of the experimental material is considered, it is assumed that the width and height of the experimental material are Unlimited. The total time step is the total simulation time length when calculating the phonon heat transport of the experimental material at the interface, and the time step is the sampling point in the total time step, that is to say, The total time step is composed of several time steps. For example: in a simulation experiment, the experiment simulates the temperature change of the experimental material from time 0 to 1000 picoseconds (ps), where the time length of 1000 ps is the total time step. According to the actual experimental needs, 1000ps is divided into 1000 sampling points, that is, one sampling is performed at 1ps, then 1ps is the first sampling point, and this sampling point is the time step. Among them, the system parameters include phonon intensity and phonon pseudo-equilibrium intensity. It should be understood that in order to avoid the divergence of calculation results and speed up the convergence speed, those skilled in the art should use the average value of phonon mean free path and relaxation time as a reference when setting the space step size and time step size. Try to make the space step and time step smaller than the average value of phonon mean free path and relaxation time.

Step S2, setting the initial value of phonon intensity and phonon pseudo-balance intensity value; this step S2 specifically includes: based on the initial value of temperature, obtaining its corresponding phonon intensity, phonon reference temperature and first deviation intensity; The initial value of the temperature is used to obtain the corresponding phonon pseudo-equilibrium intensity and the second deviation intensity.

In a specific embodiment, based on the initial value of temperature, obtaining its corresponding phonon intensity, phonon reference temperature and first deviation intensity specifically includes the following steps: Step S21, setting the initial value of temperature to T ₀ , In the case where the initial value of the known temperature is _T0 , the phonon intensity is solved based on the following formula:

In the formula, I _{ω, p} is the phonon intensity, v _g is the phonon group velocity,

is the reduced Planck constant, ω is the phonon angular frequency, p is the phonon branch, f _{ω, p} is the phonon distribution, D is the phonon density of state, D(ω, p) means that the phonon branch is p and Phonon density of states at phonon angular frequency ω.

Step S22, based on the phonon intensity, obtain the phonon reference equilibrium intensity as:

In the formula,

is the phonon reference equilibrium strength,

To consider the Bose-Einstein distribution of the constant type reference temperature T ^ref .

Step S23, based on the phonon strength and the phonon reference balance strength, obtain the first deviation strength:

In the formula, Ψ _{ω, p} is the first deviation intensity, I _{ω, p} is the phonon intensity,

Equilibrium strength for the phonon reference.

In a specific embodiment, based on the initial value of the pseudo-temperature, obtaining the corresponding phonon pseudo-balance intensity and the second deviation intensity specifically includes:

Step S24, set the initial value of the pseudo-temperature equal to the initial value of the equilibrium value as T ₀ , that is, assume that the phonon temperature and the phonon pseudo-temperature value are both T ₀ , based on the initial value of the pseudo-temperature, obtain the acoustic The sub-pseudo-balance strength is:

In the formula,

is the phonon pseudo-equilibrium intensity,

is the Bose-Einstein distribution considering the initial pseudo-temperature T ^pse ;

Step S25, based on the phonon strength and the phonon pseudo-balance strength, obtain the second deviation strength:

In the formula,

is the second bias strength.

Step S3, obtaining the first phonon intensity based on the linear Boltzmann equation. That is, the phonon intensity acquired in step S21 is iteratively updated based on the linear Boltzmann equation, and the acquired iteratively updated phonon intensity is defined as the first phonon intensity.

In a specific embodiment, obtaining the first phonon intensity based on the Boltzmann equation specifically includes: the linear Boltzmann equation is:

In the formula, Ψ _{ω, p} is the first deviation intensity,

is the second deviation strength, τ _{ω, p} is the phonon relaxation time, ω is the phonon angular frequency, p is the phonon branch, v _g is the phonon group velocity, v _g (ω, p) is the phonon branch as The phonon group velocity of p and phonon angular frequency ω. It should be understood that this application is based on the characteristics of the space-time discrete coordinate method, and the combined phonon distribution not only depends on the spatial coordinates, but also depends on the time coordinates. Therefore, the obtained phonon intensity is iteratively updated based on the linear Boltzmann equation. Specifically, the differential format is used for both time and space differentiation. In this way, not only the variation law of system temperature and heat flow along the spatial coordinates can be obtained, but also the evolution law of the above-mentioned macroscopic information changing with time can be obtained. Because in this application, the total calculation time is divided into several time steps, and the thermal transport of phonons at the interface is solved based on any time step, it is necessary to consider the mutual exchange of phonon information at the interface situation, that is to say, in the actual experimental process, the interface conditions need to be considered, and the interface conditions are:

In the formula, A and B represent two materials respectively, φ is the angle between the interface normal vector and the group velocity, the positive direction of the interface normal is from material A to material B, α _AB (ω) and α _BA (ω) are respectively Spectrum interface penetration coefficient from material A to material B and material B to material A, Ψ _{B, ω, p} is the first deviation intensity of material B, Ψ _{A, ω, p} is the first deviation intensity of material A. It should be understood that two adjacent materials are used as an example for illustration here. In actual situations, even if the same material has multiple materials different from its adjacent materials, those skilled in the art can make any Two adjacent materials are grouped to consider their interface conditions.

Step S4, obtaining temperature, pseudo temperature and heat flow based on the first phonon intensity. Specifically include the following steps:

Step S31, based on the basic definition of phonon deviation intensity and temperature, obtain the linear expression of temperature:

In the formula,

To consider the Bose-Einstein distribution at temperature T, T ^ref is a constant reference temperature, θ is a polar coordinate,

is the azimuth angle, ω _{max, p} represents the maximum phonon angular frequency of the phonon branch p, v _g is the phonon group velocity, Ψ _{ω, p} is the first deviation strength,

is the spectral volume heat capacity, that is, the volume heat capacity of each phonon branch and unit phonon angle frequency range;

Among them, the basic definition of temperature is:

Step S32, based on the basic definition of phonon deviation strength and pseudo temperature, obtain the linear expression of pseudo temperature:

In the formula, in the formula: T ^pse is the pseudo temperature, τ _{ω, p} is the phonon relaxation time;

Among them, the basic definition of pseudo temperature is:

Step S33, based on the phonon deviation intensity, obtain the linear expression of the heat flow:

In the formula, q is heat flow.

Obtain the difference between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and define the difference as a relative error;

Step S5, obtaining the relative error of two similar temperatures. Specifically include the following steps:

Step S51, obtain the relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:

In the formula, ε represents the relative error, n represents the spatial unit index; N represents the total number of spatial units; i represents the iteration number index, and T is the temperature;

Step S52, if the relative error is not greater than the convergence error, output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity, and continue to judge whether the calculation time is greater than or equal to the final evolution time. That is, it is judged whether the current calculation time exceeds the preset total calculation time.

Specifically, in the above step S52, in response to the relative error being not greater than the convergence error, output the temperature, pseudo temperature, and heat flow corresponding to the current phonon deviation intensity, and continue to judge whether the calculation time is greater than or equal to the final evolution time. It should be understood that the output temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity here are the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity output at the current time step. Because the total calculation time is composed of multiple time steps, the purpose of judging whether the current calculation time exceeds the preset total calculation time is to proceed to the next time step if the current calculation time does not exceed the preset total calculation time Calculation of long temperature, pseudo temperature and heat flow.

In a specific embodiment, in step S5, the relative error of two close temperatures is acquired. Specifically, the following steps may also be included:

Step S53. Obtain the difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and define the difference as a relative error:

In the formula, ε is the relative error; n is the spatial unit index; N is the total number of spatial units; i is the iteration number index, and q is the heat flow;

Step S54, compare the relative error with the preset convergence error, if the relative error is not greater than the convergence error, output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity, and continue to judge whether the calculation time is greater than or equal to the final evolution time . That is, it is judged whether the current calculation time exceeds the preset total calculation time.

Specifically, in the above step S54, in response to the relative error being not greater than the convergence error, output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity, and continue to judge whether the calculation time is greater than or equal to the final evolution time. It should be understood that the output temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity here are the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity output at the current time step. Because the total calculation time is composed of multiple time steps, the purpose of judging whether the current calculation time exceeds the preset total calculation time is to proceed to the next time step if the current calculation time does not exceed the preset total calculation time Calculation of long temperature, pseudo temperature and heat flow. It should be understood that in actual application scenarios, those skilled in the art can choose to obtain the difference between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity through steps S51 and S52, or Obtain the difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity through steps S53 and S54, and compare the obtained difference with the preset convergence error to determine whether the result is fulfil requirements. Similarly, those skilled in the art can also choose to compare the difference between the current heat flow and the previous heat flow and the difference between the current temperature and the previous temperature in an experiment with the preset convergence values respectively set corresponding to the above two. Errors are compared to ensure that the results meet actual requirements.

In a specific embodiment, continuing to judge whether the calculation time is greater than or equal to the final evolution time specifically includes the following steps:

Step S6, obtaining the calculation duration for iteratively updating the phonon deviation strength, and defining the calculation duration as the first calculation duration;

Step S7. If the first calculation duration is less than the preset total calculation duration, continue to execute steps S3, S4, S5, S6 and S7; otherwise, output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity.

Specifically, in the above step S7, in response to the first calculation duration being less than the preset total calculation duration, continue to execute steps S3, S4, S5, S6 and S7; in response to the first calculation duration being not less than the preset total calculation duration , output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation strength. It should be understood that the embodiment so far is a process of solving the heat transport of phonons at the interface with one time step. If the first calculation time length is less than the preset total calculation time length, then continue to execute steps S3, S4, S5, S6 and S7, that is, to solve the heat transport of phonons at the phonons of the next time step; otherwise, if If the first calculation duration is greater than or equal to the preset total calculation duration, the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity are output. Moreover, it should be understood that if the first calculation duration is less than the preset total calculation duration, then continue to execute steps S3, S4, S5, S6 and S7, that is, perform the next iterative update of the first phonon intensity, that is, the current The first phonon intensity is brought into step S3, and the current first phonon intensity is iteratively updated to obtain the first phonon intensity again until the first calculation duration is greater than or equal to the preset total calculation duration. It should be understood that in this embodiment, no specific values are given for the preset setting parameters and system parameters such as convergence error, total calculation time, and time step. Those skilled in the art can Reasonable selection according to the situation, only need to meet the relevant requirements in this embodiment.

This embodiment can directly, accurately and efficiently solve the heat transport of phonons at the inner interface of the semiconductor, and ensure the accuracy of the solution results to the greatest extent.

Embodiment two:

Referring to FIG. 3 , FIG. 3 is a flow chart of the method in the second embodiment.

The method for solving the heat transport of phonons at the interface in this embodiment includes the following steps: step S1, initialize setting parameters and system parameters; step S2, obtain the phonon deviation intensity based on the system parameters; step S3, based on the linear Bohr Zeman equation and phonon interface conditions, update the phonon deviation strength iteratively, and obtain the corresponding temperature, pseudo temperature and heat flow based on any phonon deviation strength; step S4, obtain the temperature corresponding to the current phonon deviation strength and the previous phonon deviation strength The temperature difference corresponding to the deviation intensity, and define the difference as a relative error; step S5, compare the relative error with the preset convergence error, if the relative error is not greater than the convergence error, then output the current phonon deviation intensity corresponding to Temperature, Pseudo-Temperature and Heat Flow. Specifically, in the above step S5, in response to the fact that the relative error is not greater than the convergence error, the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity are output.

In one embodiment, the method further includes: step S6, obtaining the calculation duration for iteratively updating the phonon deviation strength, and defining the calculation duration as the first calculation duration; step S7, if the first calculation duration is less than the preset total calculation duration, Then continue to execute steps S3, S4, S5, S6 and S7; otherwise, output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity. Specifically, in the above step S7, in response to the first calculation duration being less than the preset total calculation duration, continue to execute steps S3, S4, S5, S6 and S7; in response to the first calculation duration being not less than the preset total calculation duration , output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation strength.

In one embodiment, the setting parameters include: space step size, system scale, time step size, total calculation time and convergence error; system parameters include: temperature and pseudo temperature; phonon deviation strength includes: obtained based on phonon strength The first bias strength and the second bias strength obtained based on the phonon pseudo-equilibrium strength.

In one embodiment, step S2 specifically includes: step S21, the initial value of the set temperature is T ₀ , based on the initial value of the temperature, the obtained phonon intensity is:

In the formula, I _{ω, p} is the phonon intensity, v _g is the phonon group velocity,

is the reduced Planck constant, ω is the phonon angular frequency, p is the phonon branch, f _{ω, p} is the phonon distribution, D is the phonon density of state, D(ω, p) means that the phonon branch is p and Phonon density of states with phonon angular frequency ω;

Step S22, based on the phonon intensity, obtain the phonon reference equilibrium intensity as:

In the formula,

is the phonon reference equilibrium strength,

To consider the Bose-Einstein distribution of the constant type reference temperature T ^ref ;

Step S23, based on the phonon strength and the phonon reference balance strength, obtain the first deviation strength:

In the formula, Ψ _{ω, p} is the first deviation intensity, I _{ω, p} is the phonon intensity,

is the phonon reference equilibrium strength;

Step S24, set the initial value of the pseudo-temperature equal to the initial value of the equilibrium value as T ₀ , based on the initial value of the pseudo-temperature, obtain the pseudo-equilibrium strength of the phonon as:

In the formula,

is the phonon pseudo-equilibrium intensity,

is the Bose-Einstein distribution considering the initial pseudo-temperature T ^pse ;

Step S25, based on the phonon strength and the phonon pseudo-balance strength, obtain the second deviation strength:

In the formula,

is the second bias strength.

In one embodiment, the linear Boltzmann equation is:

In the formula, Ψ _{ω, p} is the first deviation intensity,

is the second deviation strength, τ _{ω, p} is the phonon relaxation time, ω is the phonon angular frequency, p is the phonon branch, v _g is the phonon group velocity, v _g (ω, p) is the phonon branch as The phonon group velocity of p and phonon angular frequency ω;

The interface conditions are:

In the formula, A and B represent two materials respectively, φ is the angle between the interface normal vector and the group velocity, the positive direction of the interface normal is from material A to material B, α _AB (ω) and α _BA (ω) are respectively Spectrum interface penetration coefficient from material A to material B and material B to material A, Ψ _{B, ω, p} is the first deviation intensity of material B, Ψ _{A, ω, p} is the first deviation intensity of material A.

In one embodiment, obtaining the corresponding temperature, pseudo temperature and heat flow based on any phonon deviation intensity specifically includes: step S31, based on the basic definition of phonon deviation intensity and temperature, obtaining a linear expression of temperature:

In the formula,

To consider the Bose-Einstein distribution at temperature T, T ^ref is a constant reference temperature, θ is a polar coordinate,

is the azimuth angle, ω _{max, p} represents the maximum phonon angular frequency of the phonon branch p, v _g is the phonon group velocity, Ψ _{ω, p} is the first deviation strength,

is the spectral volume heat capacity, that is, the volume heat capacity of each phonon branch and unit phonon angle frequency range;

Among them, the basic definition of temperature is:

Step S32, based on the basic definition of phonon deviation strength and pseudo temperature, obtain the linear expression of pseudo temperature:

In the formula: T ^pse is the pseudo temperature, τ _{ω, p} is the phonon relaxation time;

Among them, the basic definition of pseudo temperature is:

Step S32, the basic definition of pseudo temperature is:

Step S33, based on the phonon deviation intensity, obtain the linear expression of the heat flow:

In the formula, q is heat flow.

In one embodiment, step S5 specifically includes: step S51, obtaining the relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:

In the formula, ε represents the relative error, n represents the spatial unit index; N represents the total number of spatial units; i represents the iteration number index, T is the temperature; step S52, if the relative error is not greater than the convergence error, then output the current phonon deviation intensity corresponding to Temperature, Pseudo-Temperature and Heat Flow.

In one embodiment, step S5 specifically includes:

Step S53. Obtain the difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and define the difference as a relative error:

In the formula, ε is the relative error; n represents the spatial unit index; N represents the total number of spatial units; i represents the iteration number index, and q is the heat flow; step S54, compare the relative error with the preset convergence error, if the relative error is not greater than Convergence error, the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation strength are output.

In one of the embodiments, obtaining the corresponding temperature, pseudo temperature, and heat flow based on any phonon deviation intensity includes: obtaining the temperature corresponding to any phonon deviation intensity based on the basically defined ratio between any phonon deviation intensity and temperature; The ratio of any phonon deviation intensity to the basic definition of pseudo temperature is used to obtain the pseudo temperature corresponding to any phonon deviation intensity; the heat flow corresponding to any phonon deviation intensity is obtained by integral summation based on any phonon deviation intensity.

Embodiment three:

In a specific embodiment, a double-layer thin film is taken as an example, and its one-dimensional unsteady state phonon heat transport is solved. Specifically, the bilayer film is composed of germanium and silicon with equal thickness, and the film thickness is 120 nanometers. Among them, as shown in Figure 4, the two ends of the film are isothermal boundaries, which are fixed at 301K and 299K respectively, which meet the requirements of the small temperature difference between the two sides of the interface in this application. The initial temperature inside the film is a uniform temperature of 300K, and the temperature and heat flow gradually tend to steady state. The temperature evolution diagram and heat flow evolution diagram of the above-mentioned double-layer thin film at multiple different time steps obtained based on the Monte Carlo method are used as a reference, based on the traditional method for solving the phonon heat transport at the interface and the solution interface proposed by this application The method of phonon heat transport is used to verify the accuracy of the two methods.

In a specific embodiment, as shown in Figure 5, that is, based on the double-layer film shown in Figure 4, the above-mentioned double-layer thin film obtained based on the prior art, the method of the present application and the Monte Carlo method obtained under set parameters A plot of the temperature evolution of the thin film at several different time steps. Specifically include: the initial space step is 0.2 nanometers, the time step is 0.2 ps, the total calculation time is 0.6 nanoseconds, the convergence error is 1*10 ^-8 , and the initial value of the temperature is 300 Kelvin. Among them, the figure includes the temperature evolution curve of the double-layer film with a time step of 6ps, a time step of 48ps and a time step of 600ps, and also includes the method based on the application under any of the above-mentioned time steps , Based on the traditional method of solving the phonon heat transport at the interface and the temperature evolution curve of the above-mentioned double-layer thin film obtained based on the Monte Carlo method, the temperature curves obtained by the three methods at any time step are displayed together , for comparison. As can be seen from Fig. 5, the curve of temperature calculated by the traditional method is quite different from the correct value obtained based on the Monte Carlo method, while the curve of temperature obtained by the method of the present application is different from that obtained based on the Monte Carlo method. The difference between the correct values of is small, which proves that the method for solving the phonon heat transport at the interface of the present application can guarantee the accuracy and precision of the calculation results to the greatest extent.

In one of the embodiments, as shown in Figure 6, that is, based on the double-layer film shown in Figure 4, the above-mentioned double-layer thin film obtained based on the prior art, the method of the present application and the Monte Carlo method obtained under set parameters Heat flow evolution plot of a thin film at several different time steps. Wherein, the initialization space step is 0.2 nanometers, the time step is 0.2 ps, the total calculation time is 0.6 nanoseconds, the convergence error is 1*10 ^-8 , and the initial value of the temperature is 300 Kelvin. Wherein, Fig. 6 includes the heat flow evolution curve of the double-layer thin film with a time step of 6ps, a time step of 48ps and a time step of 600ps, and also includes the heat flow evolution curve based on the application under any of the above-mentioned time steps. method, based on the traditional method of solving the heat transport of phonons at the interface and the heat flow evolution curve of the above-mentioned double-layer film obtained based on the Monte Carlo method, the heat flow curves obtained by the three methods at any time step are shown in together for comparison. As can be seen from Figure 6, the heat flow curve calculated by the traditional method is quite different from the correct value obtained based on the Monte Carlo method, while the heat flow curve obtained by the method of the present application is different from that obtained based on the Monte Carlo method. The difference between the correct values of is small, which proves that the method for solving the phonon heat transport at the interface of the present application can guarantee the accuracy and precision of the calculation results to the greatest extent. It should be understood that in both Figure 5 and Figure 6, the X-axis represents the dimensionless space coordinate, specifically the ratio between the space coordinate and the film thickness; the Y-axis of Figure 5 represents the temperature, and the unit is Kelvin; the Y-axis of Figure 6 Indicates heat flow in watts per square meter.

Embodiment four:

This embodiment provides one or more non-volatile computer-readable storage media storing computer-readable instructions. When the computer-readable instructions are executed by one or more processors, the one or more processors execute any one of the above-mentioned The steps of the method for solving the phonon heat transport at the interface provided in the embodiment.

Those skilled in the art should understand that the embodiments in the embodiments of the present application may be provided as methods, systems, or computer-readable instruction products. Therefore, the embodiment of the present application may be in the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the embodiments of the present application may adopt computer-readable instruction products implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes therein. form.

Embodiments of the present application are described with reference to flowcharts and/or block diagrams of methods, devices (systems), and computer-readable instruction products according to embodiments of the present application. It should be understood that each process and/or block in the flowchart and/or block diagram, and a combination of processes and/or blocks in the flowchart and/or block diagram can be implemented by computer-readable instructions. These computer-readable instructions can be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing device to produce a machine such that the instructions executed by the processor of the computer or other programmable data processing device Produce means for realizing the functions specified in one or more procedures of the flowchart and/or one or more blocks of the block diagram.

These computer-readable instructions instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing apparatus to operate in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture comprising instruction means, The instruction means implements the functions specified in one or more procedures of the flowchart and/or one or more blocks of the block diagram.

These computer-readable instructions instructions can also be loaded on a computer or other programmable data processing device, so that a series of operational steps are performed on the computer or other programmable device to produce a computer-implemented process, thereby The executed instructions provide steps for implementing the functions specified in the procedure or procedures of the flowchart and/or the block or blocks of the block diagrams.

Note that the above are only preferred embodiments and technical principles used in this application. Those skilled in the art will understand that the present application is not limited to the specific embodiments described herein, and various obvious changes, readjustments and substitutions can be made by those skilled in the art without departing from the protection scope of the present application. Therefore, although the present application has been described in detail through the above embodiments, the present application is not limited to the above embodiments, and can also include more other equivalent embodiments without departing from the concept of the present application. The scope is determined by the scope of the appended claims.

Claims (11)

1. A method for solving phonon heat transport at an interface, characterized in that: the method comprises the following steps:

   Step S1, initializing setting parameters and system parameters;

   Step S2, based on the system parameters, obtain the phonon deviation intensity;

   Step S3, based on the linear Boltzmann equation and phonon interface conditions, iteratively updating the phonon deviation strength, and obtaining the corresponding temperature, pseudo temperature and heat flow based on any of the phonon deviation strengths;

   Step S4, obtaining the difference between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity, and defining the difference as a relative error; and

   Step S5 , comparing the relative error with a preset convergence error, and if the relative error is not greater than the convergence error, then output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity.
2. The method for solving the heat transport of phonons at the interface according to claim 1, wherein the method further comprises:

   Step S6, obtaining the calculation duration for iteratively updating the phonon deviation strength, and defining the calculation duration as the first calculation duration; and

   Step S7. If the first calculation duration is less than the preset total calculation duration, continue to execute steps S3, S4, S5, S6 and S7; otherwise, output the temperature corresponding to the current phonon deviation intensity, pseudo temperature and heat flow.
3. The method for solving the heat transport of phonons at the interface according to claim 1 or 2, characterized in that:

   The setting parameters include: space step size, system scale, time step size, total calculation time and convergence error;

   The system parameters include: temperature and pseudo temperature;

   The phonon deviation intensity includes: a first deviation intensity acquired based on phonon intensity and a second deviation intensity acquired based on phonon pseudo-balance intensity.
4. The method for solving the heat transport of phonons at the interface according to claim 3, characterized in that: the step S2 includes obtaining the first deviation intensity based on the phonon intensity;

   The acquiring and obtaining the first deviation intensity specifically includes the following steps:

   Step S21, setting the initial value of the temperature as T ₀ , based on the initial value of the temperature, the obtained phonon intensity is:

   

   In the formula, I _{ω, p} is the phonon intensity, v _g is the phonon group velocity,

   

   is the reduced Planck constant, ω is the phonon angular frequency, p is the phonon branch, f _{ω, p} is the phonon distribution, D is the phonon density of state, D(ω, p) means that the phonon branch is p and Phonon density of states with phonon angular frequency ω;

   Step S22, based on the phonon intensity, obtain the phonon reference equilibrium intensity as:

   

   In the formula,

   

   is the phonon reference equilibrium strength,

   

   To consider the Bose-Einstein distribution of the constant type reference temperature T ^ref ;

   And, step S23, based on the phonon strength and the phonon reference balance strength, obtain the first deviation strength:

   

   In the formula, Ψ _{ω, p} is the first deviation intensity, I _{ω, p} is the phonon intensity,

   

   Equilibrium strength for the phonon reference.
5. The method for solving the heat transport of phonons at the interface according to claim 4, characterized in that: the step S2 also includes obtaining the second deviation intensity based on the pseudo-equilibrium intensity of phonons;

   The acquiring and obtaining the second deviation intensity specifically includes the following steps:

   Step S24, setting the initial value of the pseudo-temperature equal to the initial value of the equilibrium value as T ₀ , based on the initial value of the pseudo-temperature, the obtained phonon pseudo-equilibrium strength is:

   

   In the formula,

   

   is the phonon pseudo-equilibrium intensity,

   

   is the Bose-Einstein distribution considering the initial pseudo-temperature T ^pse ;

   And, step S25, based on the phonon strength and the phonon pseudo-balance strength, obtaining a second deviation strength:

   

   In the formula,

   

   is the second bias strength.
6. The method for solving the heat transport of phonons at the interface according to claim 5, characterized in that:

   The linear Boltzmann equation is:

   

   In the formula, Ψ _{ω, p} is the first deviation intensity,

   

   is the second deviation strength, τ _{ω, p} is the phonon relaxation time, ω is the phonon angular frequency, p is the phonon branch, v _g is the phonon group velocity, v _g (ω, p) is the phonon branch as The phonon group velocity of p and phonon angular frequency ω;

   The interface conditions are:

   

   

   In the formula, A and B represent two materials respectively, φ is the angle between the interface normal vector and the group velocity, the positive direction of the interface normal is from material A to material B, α _AB (ω0 and α _BA (ω) are respectively from Spectral interface penetration coefficients from material A to material B and from material B to material A, Ψ _{B, ω, p} are the first deviation strengths of material B, and Ψ _{A, ω, p} are the first deviation strengths of material A.
7. The method for solving the heat transport of phonons at the interface according to claim 6, wherein the obtaining of the corresponding temperature, pseudo temperature and heat flow based on any phonon deviation intensity specifically includes:

   Step S31, based on the basic definition of the phonon deviation intensity and temperature, obtain the linear expression of temperature:

   

   In the formula,

   

   To consider the Bose-Einstein distribution at temperature T, T ^ref is a constant reference temperature, θ is a polar coordinate,

   

   is the azimuth angle, ω _{max, p} represents the maximum phonon angular frequency of the phonon branch p, v _g is the phonon group velocity, Ψ _{ω, p} is the first deviation strength,

   

   is the spectral volume heat capacity, that is, the volume heat capacity of each phonon branch and unit phonon angle frequency range;

   Among them, the basic definition of temperature is:

   

   Step S32, based on the basic definition of the phonon deviation strength and the pseudo temperature, obtain the linear expression of the pseudo temperature:

   

   In the formula: T ^pse is the pseudo temperature, τ _{ω, p} is the phonon relaxation time;

   Wherein, the basic definition of the pseudo temperature is:

   

   And, step S33, based on the phonon deviation intensity, obtain the linear expression of the heat flow:

   

   In the formula, q is heat flow.
8. The method for solving the heat transport of phonons at the interface according to claim 7, characterized in that: the step S5 specifically includes:

   Step S51. Obtain the relative error between the temperature corresponding to the current phonon deviation intensity and the temperature corresponding to the previous phonon deviation intensity based on the following formula:

   

   In the formula, ε represents the relative error, n represents the spatial unit index; N represents the total number of spatial units; i represents the iteration number index, and T is the temperature;

   And, step S52, if the relative error is not greater than the convergence error, then output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation strength.
9. The method for solving the heat transport of phonons at the interface according to claim 7, characterized in that: the step S5 specifically includes:

   Step S53. Obtain the difference between the heat flow corresponding to the current phonon deviation intensity and the heat flow corresponding to the previous phonon deviation intensity based on the following formula, and define the difference as a relative error:

   

   In the formula, ε is the relative error; n is the spatial unit index; N is the total number of spatial units; i is the iteration number index, and q is the heat flow;

   And, step S54, comparing the relative error with the preset convergence error, if the relative error is not greater than the convergence error, then output the temperature, pseudo temperature and heat flow corresponding to the current phonon deviation intensity .
10. The method for solving phonon heat transport at an interface according to claim 1, wherein the obtaining of the corresponding temperature, pseudo temperature and heat flow based on any one of the phonon deviation strengths includes:

    Obtaining a temperature corresponding to any one of the phonon deviation strengths based on a substantially defined ratio of the phonon deviation strength to temperature;

    Obtaining a pseudo temperature corresponding to any one of the phonon deviation strengths based on a substantially defined ratio of the phonon deviation strength to a pseudo temperature;

    Integral summation is performed based on any of the phonon deviation strengths to obtain a heat flow corresponding to any of the phonon deviation strengths.
11. One or more non-transitory computer-readable storage media storing computer-readable instructions is characterized in that, when the computer-readable instructions are executed by one or more processors, the one or more processors execute The steps of the method according to any one of claims 1-10.

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